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# White Powder Gold

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colloidal gold is a black/gray color in a clear liquid (I use it in photography), Gold Chloride on the other hand is indeed Gold colored.

yes you can absorb amounts of this into your system without a problem, Im not sure if it plays any biological role though?

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Does it turn you colors like colloidal silver sometimes does?

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no but gold salts will ruin your kidneys and liver, these are the gold compounds that'll get absorbed.

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Ive no idea in all honesty, but my Opinion would be yes, it should have a Very similar effect.

I know silver nitrate will go gray/black on the skin after UV (sunshine) exposure and Gold Chloride goes Purple/black with same.

so I would say yes it should have the same effect (IMO).

edit: yes I was correct in my assessment, such a condition Does exist: "Chrysiasis"

http://en.wikipedia.org/wiki/Chrysiasis

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• 1 year later...

My apologies for bringing up an old topic but it recently became one of interest for me as I was exploring it as a possible supplement I would try.

I always find it amusing when armchair "scientists" declare with absolution that something is crap or bogus or can't be true, just because it appears to be outlandish on the surface. A true scientist or one with a scientific mind, without any agenda should allow for ANY possibility and seek to explore the validity of that possibility.

It appears that a peer reviewed paper published in 1997 no less shows the dramatic improvement in the condition of rheumatoid arthritis sufferers through the administration of colloidal gold containing monovalent gold or gold salt:

http://informahealthcare.com/doi/abs/10.1080/13590849762411

Journal of Nutritional and Environmental Medicine. Seems there is something going on here. I suspect that the claims of gold being able to repair damaged DNA may not be so outlandish. Gold is a substance used to anchor DNA strands when being examined by electron tunneling microscopes so perhaps the monovalent gold bonds to the DNA strands or helps the DNA maintain its integrity. The theory has some validity.

As for the Chrysiasis issue, that problem was more of an issue with the treatment of tuberculosis with much higher doses. And the discoloration appears around the injection site. Do you guys actually read the links you're citing?

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I'm aware of tests done on cancer cells with a gold nano particles.

http://www.jnanobiotechnology.com/content/6/1/2#IDAYBQ5M

I know individuals who think that monatomic gold brings on an enlightened state and I find them a little nutty.

Maybe just start a practice of daily meditation for that end.

Here's Dave Hudson and his two bobs worth too.

Edited by throng
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My apologies for bringing up an old topic but it recently became one of interest for me as I was exploring it as a possible supplement I would try.

I always find it amusing when armchair "scientists" declare with absolution that something is crap or bogus or can't be true, just because it appears to be outlandish on the surface. A true scientist or one with a scientific mind, without any agenda should allow for ANY possibility and seek to explore the validity of that possibility.

It appears that a peer reviewed paper published in 1997 no less shows the dramatic improvement in the condition of rheumatoid arthritis sufferers through the administration of colloidal gold containing monovalent gold or gold salt:

http://informahealthcare.com/doi/abs/10.1080/13590849762411

Journal of Nutritional and Environmental Medicine. Seems there is something going on here. I suspect that the claims of gold being able to repair damaged DNA may not be so outlandish. Gold is a substance used to anchor DNA strands when being examined by electron tunneling microscopes so perhaps the monovalent gold bonds to the DNA strands or helps the DNA maintain its integrity. The theory has some validity.

As for the Chrysiasis issue, that problem was more of an issue with the treatment of tuberculosis with much higher doses. And the discoloration appears around the injection site. Do you guys actually read the links you're citing?

Do you?

Gold salts of monovalent gold (AU I) with a gold-sulfur ligand (aurothiolates) are the only form of gold currently in use for the management of rheumatoid arthritis (RA)

Gold salts. Not monoatomic gold, aka ORMUS, which is a bunch hooey.

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• 1 month later...

Dear travis418 and all the contributors of this discussion.

Please know that I have an engineering degree, so I consider this as an educated answer.

I've worked with Alchemy for quite some time now, and I feel compelled to answer your questions from the perspective of my own experience with Alchemy (not from speculation).

I understand this discussion might be controversial to many, since we are on a science forum. Please understand that science offers simply a model of reality as currently perceived, true scientists know our understanding evolved over time. Their attitude is of humble curiosity and they are rarely infected by the illusion that they know it all, and by the arrogance that comes with it.

I seek not to convince anyone, especially the many "know it all" that are probably deeply rooted in classical materialism - I seek to answer those who genuinely want to know.

My experience comes from being taught Alchemy by those guys www.treasurealchemy.com

First of all, the monoatomic/ORMUS/ORMEs/m-state people claim (and genuinely believe) they are dealing with alchemy, they are not. This is just baby stuff.

If you investigate this domain long enough, like I did, you will reach the true product of Alchemy which is a mysterious white powder that has been called the Philosopher's Stone. This is what is described in ancient texts from all traditions. It was called Mana in the Jewish tradition, Mfkzt in ancient Egypt, Shem-an-na in ancient Babylon. One can also find the following names: The Elixir of Life – Star Fire – The food of the gods – the Philosopher Stone.

This powder is real, tangible, the method of production is known (I am using it) - and you can read a bit of my journey here: http://www.realphilosopherstone.com/

Now, how can one understand this powder from a scientific perspective?

Here is a letter written by my teachers a while ago. It is available nowhere else, and it will help you understand this wacky world.

I have adapted the text a bit with links to reference sites so that you can verify some claims.

Letter:

Neither Hudson, nor anybody else we know did understand really what was happening. Chemistry deluded them. If you are totally rooted in classical chemistry and physics, your eyes will se what you believe in, you will find proof for that in analytical results, and you will not think “beyond”. You have to accept that there might be something else or something more to see that something.

We have a close friend in the US with firm chemical, physical and biological academic background who made analyses of the powder. The analyses gave quite similar results as Hudson’s, but a different method was used.

But: Hudson says, this white powder consists of monoatomic transient elements. The spectroscopic method he used is based on heating the powder for a relatively long time on a graphite electrode and then assaying it. Then it assays as these elements. Nobody ever proved that they are really monoatomic. It is just a speculation, and has been shown to be incorrect see here. Hudson thought his white powder WAS a monoatomic transient metal mixture when he put it onto the electrode. It was not!! When this powder meets graphite on high temperature and long enough time, it TURNS into these transient elements. White powder IS NOT a transient metal mixture, it TURNS to that in certain circumstances!

We put our powder into a graphite vessel, covered it all with graphite powder and then heated it at 1400 Celsius for 24-36 hours. Then the tiny metallic particles produced during this time and the remaining powder were assayed – and guess what, the mixture assayed as silica, gold, copper, magnesium, potassium, chromium, iron, sulphur, etc. When it is done in an inert atmosphere and not just under air, the results are somewhat different. The result of this method somewhat changes depending on reaction time and temperature. In some cases, the reduced powder contained some rhodium, platinum, etc.

So it turned out, that our powder transforms into a mixture of NEW elements when heating under such conditions.

As you most certainly know, SiO2 cannot be altered into any other element without extremely large energy input, requiring energy levels of an atomic reactor. When you have to jiggle electrons, protons, neutrons, you need those energy levels. Now in our case with some graphite and heat for some time, the miracle happens. So as a conclusive remark, we should add that this SiO2 is a very peculiar one, because such changes are impossible according to the present chemical and physical knowledge.

The carbon energy reduction might be what actually happens to the ingested powder in the body. The human body is some kind of an alchemical crucible. It can “use” this powder. We think something very similar is happening in our bodies but unfortunately scientifically it cannot be checked yet. However, judging subjectively from the resultant robust health and very high energy levels, the changes are tremendously beneficial.

If or when you are “ready”, it will come to you. You cannot search for it. You can only find it. Or does it find you? Maybe.

The only one real personal report on the Internet about the effect of wpg was an interview with an unidentified man from Hudson’s group. It is rather dim or misty when you read it and you don’t understand what he says and why he says it so. Now we know. Our whole experience-mass is so “otherworldly” in a way that it is almost incommunicable.

Moreover, when you put the powder into a Faraday cage, you can feel its Meisner field. We tried several methods to utilize its power. You can put it into a metal pendant and wear it in your neck, for example. It has a profound effect even this way, just by constantly interacting with the aura of the body.

There was a thermogravimetric analysis done with our powder. First, it was attempted under helium atmosphere. The powder was taken and put on a precise scale and heated up to ca. 1000 degrees Centigarde. The assay had to be terminated about halfway, because the powder behaved in a very unusual way. It's weight was jumping up and down and making the equipment and the personnel go mental. Then the next measurement was done in nitrogen atmosphere. Here the powder was calm. From 100 up to 800 degrees C, it was loosing its weight gradually and at ca. 800 it stabilised having lost 25% of its original weight. Up to 1100 degrees it did not change any more. When it cooled, its weight remained minus 25%.

Later in the letter, they wrote:

Now we can make healing powders designed exactly for a sick person to heal him/her. And it works. The only exception might be when the soul does really want to quit...

The question on how healing occurs is discussed here: http://www.treasurealchemy.com/1-creations-and-effects-of-achemy - it is discussed around 13 minutes 30 sec.

If you have questions, ask and I will do my best to answer you.

All the best to you.

Edited by Saint Germain
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If you have questions, ask and I will do my best to answer you.
I find this part a bit too convenient:

This knowledge of this powder is somewhat of an alchemical secret. Nobody talks about it. We can understand that now, this is wise to behave so. If or when you are “ready”, it will come to you. You cannot search for it. You can only find it. Or does it find you? Maybe.
It's a secret that you now believe it is wise to protect, and nobody else can search for it until they are ready to accept that it's true? If you want acceptance from the scientific community, doesn't it make sense to work within the scientific method and make some of the powder and your test results available to others so your work can be reproduced and verified, then peer reviewed?
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The word "secret" here isn't meant in the sense "needs to be kept away from others" - it is more like a realization that comes to you when your consciousness is ready.

The very idea of such an exotic material is so beyond what the average scientist has been trained to think and believe that it is automatically rejected as "nonsense".

And it IS nonsense, yet this is how it is.

As stated, I don't seek acceptance from the scientific community at all. This stuff doesn't fit in today's scientific framework, and science will need to go a long way to integrate this, if it ever can.

Nevertheless, Alchemy is a precise science (please don't choke) in the sense that it is repeatable by anyone (obviously, since I have followed my teacher's method and have produced the Stone myself), and it works every time. Now can the process be understood and explained within the current scientific understanding? Certainly not, and it might never be fully explained by science.

And if you think that the nature of the powder is weird, let me assure you that its creation violates the very foundation of physics - but I will not disclose more about this publicly.

As an Engineer, I am pleased to see that tests have been performed, that the powder has been studied with various scientific equipments and methods.

But as an Engineer, I am also well aware of the limits of science. Not everything can be fully grasped within the scientific perspective, especially the way science has been defined since Descartes, Newton, and all these mighty figures: as an objective reality made of dry impersonal matter.

As an Alchemist, I am pleased that these tests confirm that the substance is exotic - but I know the real value is in its use, not its study.

Doesn't it make sense to work within the scientific method and make some of the powder and your test results available to others so your work can be reproduced and verified, then peer reviewed?

Your question is very natural from a scientific logic.

Now let me ask you: how do you suggest to set up a thorough study of the powder using the scientific method? What would be the purpose of having it published and peer reviewed?

Let me remind you that it HAS been analyzed and the studies tell us "what you are looking at will laugh at you - I will show up as different matters, change weight".

People have performed these scientific studies and have arrived at a point where they throw up their hands in despair, saying they cannot account for what they see.

Science likes to put thing in boxes of knowledge. Many scientists actually believe that by studying, observing and measuring something, you can actually know what it is. You may know a bit ABOUT something, but you don't know what it is.

Now you may not like what I will write next, and I do not know to what degree you are educated in the nature of reality (beyond the mechanical understanding), but this is the truth.

The Stone has been sought for years, many have invested fortunes, have risked their lives to uncover the secrets of Alchemy.

Why do you think it is so?

Because when used in the alchemical context, it is magnificent gift that allows one to accomplish miracles.

The Stone isn't to be understood as matter that you can use to build stuff. The Stone reacts to consciousness - and is to be used within a spiritual context. This is the context of Alchemy as it was used in former civilizations, and as it is used today.

I understand if people refuse to integrate this in their worldview, reject this, call this view crackpot, or any other funky names. I know I would have done so a few years earlier.

Now again, we are outside of the scientific domain - no one is asking you to believe anything - I am just sharing my perspective.

Edited by Saint Germain
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People have performed these scientific studies and have arrived at a point where they throw up their hands in despair, saying they cannot account for what they see.

So, where are the peer-reviewed studies showing what they did, and how they cannot account for it? Things that behave strangely are considered very interesting by scientists, because it points to something where they could gain new understanding.

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White powder gold is a bunch of hooey! Seriously though, like many have already said above me, it is. Like Mr. Skeptic said, let us see the scientific studies.

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My dear Mr Skeptic,

You seem to have a very idealized view of science... Actually, I think scientists have two major types of behaviors when facing "things that behave strangely".

The large majority purely and simply rejects it because it doesn't fit their understanding. They treat the thing as not possible before proven otherwise, they feel threatened by the unknown and have a very attacking attitude, and seem on a crusade to preserve their current worldview (which is often hard materialism).

Only a small group of scientists actually have an open-minded attitude and consider the phenomena - these are the geniuses who help humanity make giant leaps.

Now, I could give you something from a major US university - but I'm not sure you even know why you are asking for these studies.

So, give me your view on these questions, and I will give you something:

1. What do you mean by "peer-reviewed" study? People looking at other people's measurements? Stand-alone measurements are not peer-reviewed, scientific papers offering some kind of theory or hypothesis are. Do you understand what the process of peer-reviewing entails?

2. Then the two same questions as for Phi for all.

How do you suggest to set up a thorough study of the powder using the scientific method? What would be the purpose of having it published and peer-reviewed?

Merged post follows:

Consecutive posts merged
White powder gold is a bunch of hooey! Seriously though, like many have already said above me, it is. Like Mr. Skeptic said, let us see the scientific studies.

I'm curious, why are you asking for scientific studies if you know it is "hooey"?

Some take the position that if science doesn't give us reason to believe in something, then no good reason exists. That's simply the false assumption of scientism. Let me remind you that I seek not to prove or convince, but inform on the very first question asked.

Edited by Saint Germain
Consecutive posts merged.
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The large majority purely and simply rejects it because it doesn't fit their understanding. They treat the thing as not possible before proven otherwise, they feel threatened by the unknown and have a very attacking attitude, and seem on a crusade to preserve their current worldview (which is often hard materialism).

Alternately, the unknown can provide a convenient way to get large amounts of grant money and graduate students to do research with.

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Now let me ask you: how do you suggest to set up a thorough study of the powder using the scientific method?
Take any of your non-natural results (like the one where your powder changes into other elements), show some research, make a prediction, set up an experiment to test those predictions and review the results. Share your findings with everyone.
What would be the purpose of having it published and peer reviewed?
Validity.

Let me remind you that it HAS been analyzed and the studies tell us "what you are looking at will laugh at you - I will show up as different matters, change weight".
Can you link us to those studies?

People have performed these scientific studies and have arrived at a point where they throw up their hands in despair, saying they cannot account for what they see.
This would be HUGE news. I haven't seen anything about it. Scientists love two things almost equally about peer review: confirming results and being part of the new frontier, and proving the results flawed after thorough testing. They do NOT love throwing up their hands and saying they can't account for what they see.
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Alternately, the unknown can provide a convenient way to get large amounts of grant money and graduate students to do research with.

I agree that the unknown can be used for deceit, it has been, currently is, and most probably will be.

I'm not sure it is a justification repel other suggestions with an attacking attitude, especially from scientific people, who often consider themselves as educated.

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My dear Mr Skeptic,

You seem to have a very idealized view of science... Actually, I think scientists have two major types of behaviors when facing "things that behave strangely".

The large majority purely and simply rejects it because it doesn't fit their understanding. They treat the thing as not possible before proven otherwise, they feel threatened by the unknown and have a very attacking attitude, and seem on a crusade to preserve their current worldview (which is often hard materialism).

No real scientists do this. Anything that violates the known laws of physics is considered absurdly unlikely, unless it happens to be true -- in which case, Nobel Prize gets awarded.

Only a small group of scientists actually have an open-minded attitude and consider the phenomena - these are the geniuses who help humanity make giant leaps.

Nope, only the ones who follow the scientific method make the great leaps. The others are a bunch of crazies with no evidence (if they had evidence it would be scientific).

Now, I could give you something from a major US university - but I'm not sure you even know why you are asking for these studies.

So, give me your view on these questions, and I will give you something:

1. What do you mean by "peer-reviewed" study? People looking at other people's measurements? Stand-alone measurements are not peer-reviewed, scientific papers offering some kind of theory or hypothesis are. Do you understand what the process of peer-reviewing entails?

Of course I understand peer review. A study in evidence of what you are suggesting would detail how the substance was obtained, what was done with it, and show that the results contradict or cannot be explained by the known laws of physics. The peer review would verify that said claim is not based on a misunderstanding nor on poor experimental setup.

2. Then the two same questions as for Phi for all.

How do you suggest to set up a thorough study of the powder using the scientific method? What would be the purpose of having it published and peer-reviewed?

Because there are a bunch of crazy dumb people, and no one with any sense will believe this unless they either did it for themselves (which requires it to be published) or because they are vouched for by real scientists (peer review). The paper just has to describe what was done and the results.

I'm curious, why are you asking for scientific studies if you know it is "hooey"?

He's giving you the benefit of the doubt, on the possibility that he is one who is mistaken.

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I agree that the unknown can be used for deceit, it has been, currently is, and most probably will be.

I'm not sure it is a justification repel other suggestions with an attacking attitude, especially from scientific people, who often consider themselves as educated.

I didn't say anything about deceit. I suggested that a scientist would love the unknown because it gives him something to research and can get him grant money to do the research with. Researching stuff that's already known won't get you a grant or a job -- it's pointless.

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I didn't say anything about deceit. I suggested that a scientist would love the unknown because it gives him something to research and can get him grant money to do the research with. Researching stuff that's already known won't get you a grant or a job -- it's pointless.

My apologies, I must have misunderstood you.

I agree with this mindset loving the unknown, but the shortcut to the research is too fast in my opinion.

A good example of this is psy phenomena. Dean Radin is doing a tremendous job in providing evidences of phenomena like clairvoyance, influence of REG (random event generators) with intention, telepathy, and many such phenomena - yet he had to go to a private institute to do his research since no money is being granted to do the research.

Otherwise, let me share one of my favorite quote about the open-mindset:

"Sit down before fact as a little child, be prepared to give up every preconceived notion, follow humbly wherever and whatever abysses nature leads, or you will learn nothing." - Thomas Henry Huxley

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Saint Germain, since you seem to have misplaced your logic in the defence of "white powder gold". I will attempt to clear up your misunderstandings.

"White powder gold" is what some producers call their non-colloidal gold. Monatomic gold is another name some companies call their "white powder gold" products. White powder gold, supposedly, is a water soluble gold salt that is dissolved in water. The gold salt MUST be water soluble to create an ionic solution.

Laboratory testing of products labelled as white powder gold have determined they are really gold chloride (Chlorauric acid). Which is a water soluble gold salt.

But, gold chloride is a neuro-toxin, so if it does have amazing abilities hopefully they outweigh this fact.

Some of these companies are trying to avoid the problem of selling gold chloride (a known neuro-toxin). So they have stated that their white powder gold is gold hydroxide. The problem with that claim though is that gold hydroxide is not water soluble, therefore it does not even match the description of their original product.

The preceding was a paraphrase from http://www.purestcolloids.com/ionic-gold.php

If you do a general search on Google you will find many articles that claim this to be a scam.

Another very simple flaw with white powder gold is that when you ionise gold it turns black, not white. That is common scientific knowledge.

Furthermore, the person who has claimed to discover monatomic gold is David Icke. David pronounced himself as the "son of god" during an interview with Terry Woogan in 1991. I don't know about you but It seems funny to me that the "son of god" would need to sell monatomic gold.

http://en.wikipedia.org/wiki/David_Icke

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Take any of your non-natural results (like the one where your powder changes into other elements), show some research, make a prediction, set up an experiment to test those predictions and review the results. Share your findings with everyone.Validity.

What is a non-natural phenomena?

You here share the method - not an actual experiment set-up.

In our case, what type of prediction would you be making?

Can you link us to those studies?

I can give you references.

This would be HUGE news. I haven't seen anything about it. Scientists love two things almost equally about peer review: confirming results and being part of the new frontier, and proving the results flawed after thorough testing. They do NOT love throwing up their hands and saying they can't account for what they see.

Here you hit the bull's eye, one of the greatest benefit of experimental replication is giving a cutting edge scientist the assurance that he/she isn't self-deluded by showing that results are not flawed.

Ok, you deserve something.

Here you have both a video that contains many references to such studies - among which a spectroscopy study performed at Cornell University.

http://www.treasurealchemy.com/further-scientific-proofs-of-its-exotic-nature

And here some text presenting other results (also this Cornell study).

http://www.levity.com/alchemy/hudson2.html

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You sited Dean Radin doing an amazing job by researching telepathy and such. Now I won't completely touch on the obvious physical flaws of telepathy, which include the ability of an electrical signal from a neuron in the brain being able to transfer through the skull, through the air, and then through the receiver's skull all without the connection of synapses or the transfer of the chemical neurotransmitters.

But Dean Radin works for IONS (Institute of Noetic Sciences) has been reviewed by the non-profit organisation Quackwatch. Quackwatch has stated they view IONS with considerable distrust because of wild scientific claims which have not even been submitted for peer-review. Suggesting that if these scientific studies were accurate and factually sound they would submit them for peer-review. Obviously another less than credible source.

http://en.wikipedia.org/wiki/Institute_of_Noetic_Sciences

But I don't know if you consider Wikipedia a credible source, because David Icke has publicly criticised Wikipedia as being part of the conspiracy to cover up the truth.

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Saint Germain, since you seem to have misplaced your logic in the defence of "white powder gold". I will attempt to clear up your misunderstandings.

As explained several times above, I do not intend to defend anything, I simply offer my perspective.

Also, I answered to this question: I was wondering if anyone knows about white powder gold(monoatomic gold) or any other M-state elements. - which is exactly what I am doing.

"White powder gold" is what some producers call their non-colloidal gold. Monatomic gold is another name some companies call their "white powder gold" products. White powder gold, supposedly, is a water soluble gold salt that is dissolved in water. The gold salt MUST be water soluble to create an ionic solution.

Laboratory testing of products labelled as white powder gold have determined they are really gold chloride (Chlorauric acid). Which is a water soluble gold salt.

But, gold chloride is a neuro-toxin, so if it does have amazing abilities hopefully they outweigh this fact.

Some of these companies are trying to avoid the problem of selling gold chloride (a known neuro-toxin). So they have stated that their white powder gold is gold hydroxide. The problem with that claim though is that gold hydroxide is not water soluble, therefore it does not even match the description of their original product.

The preceding was a paraphrase from http://www.purestcolloids.com/ionic-gold.php

It seems this company is talking about something else.

If you do a general search on Google you will find many articles that claim this to be a scam.

You can find anything and its opposite. I bet you can find articles about Elvis being alive and singing in Las Vegas as well...

Another very simple flaw with white powder gold is that when you ionise gold it turns black, not white. That is common scientific knowledge.

Furthermore, the person who has claimed to discover monatomic gold is David Icke. David pronounced himself as the "son of god" during an interview with Terry Woogan in 1991. I don't know about you but It seems funny to me that the "son of god" would need to sell monatomic gold.

http://en.wikipedia.org/wiki/David_Icke

When does he claimed to have discovered this? Pretty much all sources dealing with white powder gold acknowledge David Hudson as being the great (re)discovered of this material.

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You sited Dean Radin doing an amazing job by researching telepathy and such. Now I won't completely touch on the obvious physical flaws of telepathy

Now this is interesting.

This domain is now well-documented, and even the major skeptics have reviewed the results and acknowledge that something unusual is going on.

All that is left is ignorant-skeptics, with arguments such as "obvious flaws of telepathy".

which include the ability of an electrical signal from a neuron in the brain being able to transfer through the skull, through the air, and then through the receiver's skull all without the connection of synapses or the transfer of the chemical neurotransmitters.

Where did you get that from?

All that can be done today is speculations - to my knowledge, o such nonsense is being spoken about by Dean Radin.

But Dean Radin works for IONS (Institute of Noetic Sciences) has been reviewed by the non-profit organisation Quackwatch. Quackwatch has stated they view IONS with considerable distrust because of wild scientific claims which have not even been submitted for peer-review. Suggesting that if these scientific studies were accurate and factually sound they would submit them for peer-review. Obviously another less than credible source.

This is exactly my point.

Peer review is just a tool used by science to progress. If it were the measure of truth, no advancement could be done because the majority would permanently stand on its position.

Less than credible if you go for the uneducated majority, but guess what happens if you look at the facts and studies?

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Proportion word problems

Problem

Pamela drove her car 99 kilometers and used 9 liters of fuel. She wants to know how many kilometers left parenthesis, k, right parenthesis she can drive with 12 liters of fuel. She assumes the relationship between kilometers and fuel is proportional.
Which proportion could Pamela use to model this situation?

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# Why is the Cotangent Space of Complex Projective Space Not $U(1)$-Equivariant?

I'm looking at the cotangent bundle of $CP^{N}$ at the moment in the context of equivariance. For many reasons, it seems to me that this bundle is not $U(1)$-equivariant, or, in other words, cannot be constructed as from a representation of $U(1)$ in the standard manner (see here for example) where we consider $CP^{N}$ as the base space of the principle bundle $S^{2N}$. However, I cannot find a neat convincing argument for why this should be so.

MY ATTEMPT: I would guess that $U(1)$-equivariance would mean that the bundle could be expressed as a direct sum of line bundles, but I cannot see how to show that this is not the case (even though it seems most probable). Again I would guess that some sort of Chern argument comes in, but I don't know how to caclulate Chern classes without a connection. The only connection here I know is the Grassmannian, and at this point the whole thing just becomes a mess ....

-

If there were a representation $V$ of $U(1)$ and an isomorphism $TCP^n \cong S^{2n+1} \times_{U(1)} V$, then the tangent bundle of $S^{2n+1}$ would be the direct sum of the trivial bundle $S^{2n+1} \times V$ plus the trivial real line bundle (the vertical tangent bundle to the $S^1$-bundle. In particular, $S^{2n+1}$ is parallelizable, which implies, by a result by Adams, that $n=0,1$ or $3$. If $n=0$, then the question is void, for $n=1$, $TCP^1$ is obviously a complex line bundle.

That leaves the case of $CP^3$. If $TCP^3$ were a sum of three complex line bundles, then the total Chern class would be $(1+ax)(1+by)(1+cy)$ for some integers $a,b,c$ ($x$ is an appropriate generator of $H^2(CP^n)$. On the other hand, we know the total Chern class very well; it is $1+4x+6x^2+4 x^3$. This is discussed, without any reference to a connection, in many sources, e.g. Milnor-Stasheffs book. So you get

$$a+b+c=4, ab+bc+ac=6, abc=4.$$

It is easy that these equation do not have an integral solution (reduce modulo $2$ to conclude that $a,b,c$ have to be even).

Probably it is possible to do the higher dimensional cases in that arithmetical way, instead of nuking it with Adams' result.

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Great answer, thanks alot. Is Milnor-Stasheff's book a good place start on learning this stuff. It's not material that's central to my interests at the moment, so I'd like something that's not too heavy going - maybe even fun. –  Ago Szekeres Nov 30 '10 at 20:36
Moreover, what is the dimension of the module of vector fields of $S^7$. –  Ago Szekeres Nov 30 '10 at 21:08
One can deal with all the cases as you do for $n=3$. Just observe that $T_{P^n}$ is generated by global sections, since by the Euler sequence there is a surjection $\osum_0^n O_{P^n}(1)\to T_{P^n}$.So every line bundle that is direct summand of $T_{P^n}$ is also generated by global sections, hence it has total Chern class $1+ax$ with $a\ge 0$. Then it is easy to see that the analog of the equations given for $n=3$ have no solution. –  rita Nov 30 '10 at 21:13
Having taught several seminars on that, I can tell you that Milnor-Stasheff is a rather demanding text compared to others (like Hatchers unfinished book "Vector bundles and K-theory"). I also like to advertise the definition of Chern classes that Matthias Kreck gives here: mathi.uni-heidelberg.de/~kreck/stratifolds/DA_22_08.pdf , page 119 –  Johannes Ebert Nov 30 '10 at 21:20
@Ago: $S^7$ is parallelizable, so the dimension is $7$. –  Johannes Ebert Nov 30 '10 at 21:22
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## G = C3×C20.17D4order 480 = 25·3·5

### Direct product of C3 and C20.17D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C20.17D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — C6×Dic5 — C12×Dic5 — C3×C20.17D4
 Lower central C5 — C2×C10 — C3×C20.17D4
 Upper central C1 — C2×C6 — C6×D4

Generators and relations for C3×C20.17D4
G = < a,b,c,d | a3=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd-1=b-1, dcd-1=b10c-1 >

Subgroups: 416 in 152 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C5, C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C10, C10 [×2], C10 [×2], C12 [×2], C12 [×4], C2×C6, C2×C6 [×6], C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×6], C2×C12, C2×C12 [×4], C3×D4 [×2], C3×Q8 [×2], C22×C6 [×2], C30, C30 [×2], C30 [×2], C4.4D4, Dic10 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×2], C22×C10 [×2], C4×C12, C3×C22⋊C4 [×4], C6×D4, C6×Q8, C3×Dic5 [×4], C60 [×2], C2×C30, C2×C30 [×6], C4×Dic5, C23.D5 [×4], C2×Dic10, D4×C10, C3×C4.4D4, C3×Dic10 [×2], C6×Dic5 [×4], C2×C60, D4×C15 [×2], C22×C30 [×2], C20.17D4, C12×Dic5, C3×C23.D5 [×4], C6×Dic10, D4×C30, C3×C20.17D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], C2×D4, C4○D4 [×2], D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C4.4D4, C5⋊D4 [×2], C22×D5, C6×D4, C3×C4○D4 [×2], C6×D5 [×3], D42D5 [×2], C2×C5⋊D4, C3×C4.4D4, C3×C5⋊D4 [×2], D5×C2×C6, C20.17D4, C3×D42D5 [×2], C6×C5⋊D4, C3×C20.17D4

Smallest permutation representation of C3×C20.17D4
On 240 points
Generators in S240
(1 52 96)(2 53 97)(3 54 98)(4 55 99)(5 56 100)(6 57 81)(7 58 82)(8 59 83)(9 60 84)(10 41 85)(11 42 86)(12 43 87)(13 44 88)(14 45 89)(15 46 90)(16 47 91)(17 48 92)(18 49 93)(19 50 94)(20 51 95)(21 204 188)(22 205 189)(23 206 190)(24 207 191)(25 208 192)(26 209 193)(27 210 194)(28 211 195)(29 212 196)(30 213 197)(31 214 198)(32 215 199)(33 216 200)(34 217 181)(35 218 182)(36 219 183)(37 220 184)(38 201 185)(39 202 186)(40 203 187)(61 230 124)(62 231 125)(63 232 126)(64 233 127)(65 234 128)(66 235 129)(67 236 130)(68 237 131)(69 238 132)(70 239 133)(71 240 134)(72 221 135)(73 222 136)(74 223 137)(75 224 138)(76 225 139)(77 226 140)(78 227 121)(79 228 122)(80 229 123)(101 147 171)(102 148 172)(103 149 173)(104 150 174)(105 151 175)(106 152 176)(107 153 177)(108 154 178)(109 155 179)(110 156 180)(111 157 161)(112 158 162)(113 159 163)(114 160 164)(115 141 165)(116 142 166)(117 143 167)(118 144 168)(119 145 169)(120 146 170)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220)(221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 207 62 169)(2 216 63 178)(3 205 64 167)(4 214 65 176)(5 203 66 165)(6 212 67 174)(7 201 68 163)(8 210 69 172)(9 219 70 161)(10 208 71 170)(11 217 72 179)(12 206 73 168)(13 215 74 177)(14 204 75 166)(15 213 76 175)(16 202 77 164)(17 211 78 173)(18 220 79 162)(19 209 80 171)(20 218 61 180)(21 138 142 89)(22 127 143 98)(23 136 144 87)(24 125 145 96)(25 134 146 85)(26 123 147 94)(27 132 148 83)(28 121 149 92)(29 130 150 81)(30 139 151 90)(31 128 152 99)(32 137 153 88)(33 126 154 97)(34 135 155 86)(35 124 156 95)(36 133 157 84)(37 122 158 93)(38 131 159 82)(39 140 160 91)(40 129 141 100)(41 192 240 120)(42 181 221 109)(43 190 222 118)(44 199 223 107)(45 188 224 116)(46 197 225 105)(47 186 226 114)(48 195 227 103)(49 184 228 112)(50 193 229 101)(51 182 230 110)(52 191 231 119)(53 200 232 108)(54 189 233 117)(55 198 234 106)(56 187 235 115)(57 196 236 104)(58 185 237 113)(59 194 238 102)(60 183 239 111)
(1 202 11 212)(2 201 12 211)(3 220 13 210)(4 219 14 209)(5 218 15 208)(6 217 16 207)(7 216 17 206)(8 215 18 205)(9 214 19 204)(10 213 20 203)(21 84 31 94)(22 83 32 93)(23 82 33 92)(24 81 34 91)(25 100 35 90)(26 99 36 89)(27 98 37 88)(28 97 38 87)(29 96 39 86)(30 95 40 85)(41 197 51 187)(42 196 52 186)(43 195 53 185)(44 194 54 184)(45 193 55 183)(46 192 56 182)(47 191 57 181)(48 190 58 200)(49 189 59 199)(50 188 60 198)(61 165 71 175)(62 164 72 174)(63 163 73 173)(64 162 74 172)(65 161 75 171)(66 180 76 170)(67 179 77 169)(68 178 78 168)(69 177 79 167)(70 176 80 166)(101 234 111 224)(102 233 112 223)(103 232 113 222)(104 231 114 221)(105 230 115 240)(106 229 116 239)(107 228 117 238)(108 227 118 237)(109 226 119 236)(110 225 120 235)(121 144 131 154)(122 143 132 153)(123 142 133 152)(124 141 134 151)(125 160 135 150)(126 159 136 149)(127 158 137 148)(128 157 138 147)(129 156 139 146)(130 155 140 145)

G:=sub<Sym(240)| (1,52,96)(2,53,97)(3,54,98)(4,55,99)(5,56,100)(6,57,81)(7,58,82)(8,59,83)(9,60,84)(10,41,85)(11,42,86)(12,43,87)(13,44,88)(14,45,89)(15,46,90)(16,47,91)(17,48,92)(18,49,93)(19,50,94)(20,51,95)(21,204,188)(22,205,189)(23,206,190)(24,207,191)(25,208,192)(26,209,193)(27,210,194)(28,211,195)(29,212,196)(30,213,197)(31,214,198)(32,215,199)(33,216,200)(34,217,181)(35,218,182)(36,219,183)(37,220,184)(38,201,185)(39,202,186)(40,203,187)(61,230,124)(62,231,125)(63,232,126)(64,233,127)(65,234,128)(66,235,129)(67,236,130)(68,237,131)(69,238,132)(70,239,133)(71,240,134)(72,221,135)(73,222,136)(74,223,137)(75,224,138)(76,225,139)(77,226,140)(78,227,121)(79,228,122)(80,229,123)(101,147,171)(102,148,172)(103,149,173)(104,150,174)(105,151,175)(106,152,176)(107,153,177)(108,154,178)(109,155,179)(110,156,180)(111,157,161)(112,158,162)(113,159,163)(114,160,164)(115,141,165)(116,142,166)(117,143,167)(118,144,168)(119,145,169)(120,146,170), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,207,62,169)(2,216,63,178)(3,205,64,167)(4,214,65,176)(5,203,66,165)(6,212,67,174)(7,201,68,163)(8,210,69,172)(9,219,70,161)(10,208,71,170)(11,217,72,179)(12,206,73,168)(13,215,74,177)(14,204,75,166)(15,213,76,175)(16,202,77,164)(17,211,78,173)(18,220,79,162)(19,209,80,171)(20,218,61,180)(21,138,142,89)(22,127,143,98)(23,136,144,87)(24,125,145,96)(25,134,146,85)(26,123,147,94)(27,132,148,83)(28,121,149,92)(29,130,150,81)(30,139,151,90)(31,128,152,99)(32,137,153,88)(33,126,154,97)(34,135,155,86)(35,124,156,95)(36,133,157,84)(37,122,158,93)(38,131,159,82)(39,140,160,91)(40,129,141,100)(41,192,240,120)(42,181,221,109)(43,190,222,118)(44,199,223,107)(45,188,224,116)(46,197,225,105)(47,186,226,114)(48,195,227,103)(49,184,228,112)(50,193,229,101)(51,182,230,110)(52,191,231,119)(53,200,232,108)(54,189,233,117)(55,198,234,106)(56,187,235,115)(57,196,236,104)(58,185,237,113)(59,194,238,102)(60,183,239,111), (1,202,11,212)(2,201,12,211)(3,220,13,210)(4,219,14,209)(5,218,15,208)(6,217,16,207)(7,216,17,206)(8,215,18,205)(9,214,19,204)(10,213,20,203)(21,84,31,94)(22,83,32,93)(23,82,33,92)(24,81,34,91)(25,100,35,90)(26,99,36,89)(27,98,37,88)(28,97,38,87)(29,96,39,86)(30,95,40,85)(41,197,51,187)(42,196,52,186)(43,195,53,185)(44,194,54,184)(45,193,55,183)(46,192,56,182)(47,191,57,181)(48,190,58,200)(49,189,59,199)(50,188,60,198)(61,165,71,175)(62,164,72,174)(63,163,73,173)(64,162,74,172)(65,161,75,171)(66,180,76,170)(67,179,77,169)(68,178,78,168)(69,177,79,167)(70,176,80,166)(101,234,111,224)(102,233,112,223)(103,232,113,222)(104,231,114,221)(105,230,115,240)(106,229,116,239)(107,228,117,238)(108,227,118,237)(109,226,119,236)(110,225,120,235)(121,144,131,154)(122,143,132,153)(123,142,133,152)(124,141,134,151)(125,160,135,150)(126,159,136,149)(127,158,137,148)(128,157,138,147)(129,156,139,146)(130,155,140,145)>;

G:=Group( (1,52,96)(2,53,97)(3,54,98)(4,55,99)(5,56,100)(6,57,81)(7,58,82)(8,59,83)(9,60,84)(10,41,85)(11,42,86)(12,43,87)(13,44,88)(14,45,89)(15,46,90)(16,47,91)(17,48,92)(18,49,93)(19,50,94)(20,51,95)(21,204,188)(22,205,189)(23,206,190)(24,207,191)(25,208,192)(26,209,193)(27,210,194)(28,211,195)(29,212,196)(30,213,197)(31,214,198)(32,215,199)(33,216,200)(34,217,181)(35,218,182)(36,219,183)(37,220,184)(38,201,185)(39,202,186)(40,203,187)(61,230,124)(62,231,125)(63,232,126)(64,233,127)(65,234,128)(66,235,129)(67,236,130)(68,237,131)(69,238,132)(70,239,133)(71,240,134)(72,221,135)(73,222,136)(74,223,137)(75,224,138)(76,225,139)(77,226,140)(78,227,121)(79,228,122)(80,229,123)(101,147,171)(102,148,172)(103,149,173)(104,150,174)(105,151,175)(106,152,176)(107,153,177)(108,154,178)(109,155,179)(110,156,180)(111,157,161)(112,158,162)(113,159,163)(114,160,164)(115,141,165)(116,142,166)(117,143,167)(118,144,168)(119,145,169)(120,146,170), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,207,62,169)(2,216,63,178)(3,205,64,167)(4,214,65,176)(5,203,66,165)(6,212,67,174)(7,201,68,163)(8,210,69,172)(9,219,70,161)(10,208,71,170)(11,217,72,179)(12,206,73,168)(13,215,74,177)(14,204,75,166)(15,213,76,175)(16,202,77,164)(17,211,78,173)(18,220,79,162)(19,209,80,171)(20,218,61,180)(21,138,142,89)(22,127,143,98)(23,136,144,87)(24,125,145,96)(25,134,146,85)(26,123,147,94)(27,132,148,83)(28,121,149,92)(29,130,150,81)(30,139,151,90)(31,128,152,99)(32,137,153,88)(33,126,154,97)(34,135,155,86)(35,124,156,95)(36,133,157,84)(37,122,158,93)(38,131,159,82)(39,140,160,91)(40,129,141,100)(41,192,240,120)(42,181,221,109)(43,190,222,118)(44,199,223,107)(45,188,224,116)(46,197,225,105)(47,186,226,114)(48,195,227,103)(49,184,228,112)(50,193,229,101)(51,182,230,110)(52,191,231,119)(53,200,232,108)(54,189,233,117)(55,198,234,106)(56,187,235,115)(57,196,236,104)(58,185,237,113)(59,194,238,102)(60,183,239,111), (1,202,11,212)(2,201,12,211)(3,220,13,210)(4,219,14,209)(5,218,15,208)(6,217,16,207)(7,216,17,206)(8,215,18,205)(9,214,19,204)(10,213,20,203)(21,84,31,94)(22,83,32,93)(23,82,33,92)(24,81,34,91)(25,100,35,90)(26,99,36,89)(27,98,37,88)(28,97,38,87)(29,96,39,86)(30,95,40,85)(41,197,51,187)(42,196,52,186)(43,195,53,185)(44,194,54,184)(45,193,55,183)(46,192,56,182)(47,191,57,181)(48,190,58,200)(49,189,59,199)(50,188,60,198)(61,165,71,175)(62,164,72,174)(63,163,73,173)(64,162,74,172)(65,161,75,171)(66,180,76,170)(67,179,77,169)(68,178,78,168)(69,177,79,167)(70,176,80,166)(101,234,111,224)(102,233,112,223)(103,232,113,222)(104,231,114,221)(105,230,115,240)(106,229,116,239)(107,228,117,238)(108,227,118,237)(109,226,119,236)(110,225,120,235)(121,144,131,154)(122,143,132,153)(123,142,133,152)(124,141,134,151)(125,160,135,150)(126,159,136,149)(127,158,137,148)(128,157,138,147)(129,156,139,146)(130,155,140,145) );

G=PermutationGroup([(1,52,96),(2,53,97),(3,54,98),(4,55,99),(5,56,100),(6,57,81),(7,58,82),(8,59,83),(9,60,84),(10,41,85),(11,42,86),(12,43,87),(13,44,88),(14,45,89),(15,46,90),(16,47,91),(17,48,92),(18,49,93),(19,50,94),(20,51,95),(21,204,188),(22,205,189),(23,206,190),(24,207,191),(25,208,192),(26,209,193),(27,210,194),(28,211,195),(29,212,196),(30,213,197),(31,214,198),(32,215,199),(33,216,200),(34,217,181),(35,218,182),(36,219,183),(37,220,184),(38,201,185),(39,202,186),(40,203,187),(61,230,124),(62,231,125),(63,232,126),(64,233,127),(65,234,128),(66,235,129),(67,236,130),(68,237,131),(69,238,132),(70,239,133),(71,240,134),(72,221,135),(73,222,136),(74,223,137),(75,224,138),(76,225,139),(77,226,140),(78,227,121),(79,228,122),(80,229,123),(101,147,171),(102,148,172),(103,149,173),(104,150,174),(105,151,175),(106,152,176),(107,153,177),(108,154,178),(109,155,179),(110,156,180),(111,157,161),(112,158,162),(113,159,163),(114,160,164),(115,141,165),(116,142,166),(117,143,167),(118,144,168),(119,145,169),(120,146,170)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220),(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,207,62,169),(2,216,63,178),(3,205,64,167),(4,214,65,176),(5,203,66,165),(6,212,67,174),(7,201,68,163),(8,210,69,172),(9,219,70,161),(10,208,71,170),(11,217,72,179),(12,206,73,168),(13,215,74,177),(14,204,75,166),(15,213,76,175),(16,202,77,164),(17,211,78,173),(18,220,79,162),(19,209,80,171),(20,218,61,180),(21,138,142,89),(22,127,143,98),(23,136,144,87),(24,125,145,96),(25,134,146,85),(26,123,147,94),(27,132,148,83),(28,121,149,92),(29,130,150,81),(30,139,151,90),(31,128,152,99),(32,137,153,88),(33,126,154,97),(34,135,155,86),(35,124,156,95),(36,133,157,84),(37,122,158,93),(38,131,159,82),(39,140,160,91),(40,129,141,100),(41,192,240,120),(42,181,221,109),(43,190,222,118),(44,199,223,107),(45,188,224,116),(46,197,225,105),(47,186,226,114),(48,195,227,103),(49,184,228,112),(50,193,229,101),(51,182,230,110),(52,191,231,119),(53,200,232,108),(54,189,233,117),(55,198,234,106),(56,187,235,115),(57,196,236,104),(58,185,237,113),(59,194,238,102),(60,183,239,111)], [(1,202,11,212),(2,201,12,211),(3,220,13,210),(4,219,14,209),(5,218,15,208),(6,217,16,207),(7,216,17,206),(8,215,18,205),(9,214,19,204),(10,213,20,203),(21,84,31,94),(22,83,32,93),(23,82,33,92),(24,81,34,91),(25,100,35,90),(26,99,36,89),(27,98,37,88),(28,97,38,87),(29,96,39,86),(30,95,40,85),(41,197,51,187),(42,196,52,186),(43,195,53,185),(44,194,54,184),(45,193,55,183),(46,192,56,182),(47,191,57,181),(48,190,58,200),(49,189,59,199),(50,188,60,198),(61,165,71,175),(62,164,72,174),(63,163,73,173),(64,162,74,172),(65,161,75,171),(66,180,76,170),(67,179,77,169),(68,178,78,168),(69,177,79,167),(70,176,80,166),(101,234,111,224),(102,233,112,223),(103,232,113,222),(104,231,114,221),(105,230,115,240),(106,229,116,239),(107,228,117,238),(108,227,118,237),(109,226,119,236),(110,225,120,235),(121,144,131,154),(122,143,132,153),(123,142,133,152),(124,141,134,151),(125,160,135,150),(126,159,136,149),(127,158,137,148),(128,157,138,147),(129,156,139,146),(130,155,140,145)])

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A ··· 6F 6G 6H 6I 6J 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E ··· 12L 12M 12N 12O 12P 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 5 5 6 ··· 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 12 12 12 ··· 12 12 12 12 12 15 15 15 15 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 4 4 1 1 2 2 10 10 10 10 20 20 2 2 1 ··· 1 4 4 4 4 2 ··· 2 4 ··· 4 2 2 2 2 10 ··· 10 20 20 20 20 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D5 C4○D4 D10 D10 C3×D4 C3×D5 C5⋊D4 C3×C4○D4 C6×D5 C6×D5 C3×C5⋊D4 D4⋊2D5 C3×D4⋊2D5 kernel C3×C20.17D4 C12×Dic5 C3×C23.D5 C6×Dic10 D4×C30 C20.17D4 C4×Dic5 C23.D5 C2×Dic10 D4×C10 C60 C6×D4 C30 C2×C12 C22×C6 C20 C2×D4 C12 C10 C2×C4 C23 C4 C6 C2 # reps 1 1 4 1 1 2 2 8 2 2 2 2 4 2 4 4 4 8 8 4 8 16 4 8

Matrix representation of C3×C20.17D4 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 1 0 0 0 0 1
,
 3 0 0 0 15 41 0 0 0 0 60 59 0 0 1 1
,
 53 57 0 0 1 8 0 0 0 0 11 0 0 0 0 11
,
 53 57 0 0 31 8 0 0 0 0 50 39 0 0 0 11
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[3,15,0,0,0,41,0,0,0,0,60,1,0,0,59,1],[53,1,0,0,57,8,0,0,0,0,11,0,0,0,0,11],[53,31,0,0,57,8,0,0,0,0,50,0,0,0,39,11] >;

C3×C20.17D4 in GAP, Magma, Sage, TeX

C_3\times C_{20}._{17}D_4
% in TeX

G:=Group("C3xC20.17D4");
// GroupNames label

G:=SmallGroup(480,729);
// by ID

G=gap.SmallGroup(480,729);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,176,1598,303,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^4=1,d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^9,d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations

׿
×
𝔽


text:
# Extending drawings of complete graphs into arrangements of pseudocircles

Motivated by the successful application of geometry to proving the Harary-Hill Conjecture for "pseudolinear" drawings of K_n, we introduce "pseudospherical" drawings of graphs. A spherical drawing of a graph G is a drawing in the unit sphere 𝕊^2 in which the vertices of G are represented as points—no three on a great circle—and the edges of G are shortest-arcs in 𝕊^2 connecting pairs of vertices. Such a drawing has three properties: every edge e is contained in a simple closed curve γ_e such that the only vertices in γ_e are the ends of e; if e f, then γ_e∩γ_f has precisely two crossings; and if e f, then e intersects γ_f at most once, either a crossing or an end of e. We use these three properties to define a pseudospherical drawing of G. Our main result is that, for the complete graph, these three properties are equivalent to the same three properties but with "exactly two crossings" replaced by "at most two crossings". The proof requires a result in the geometric transversal theory of arrangements of pseudocircles. This is proved using the surprising result that the absence of special arcs (coherent spirals) in an arrangement of simple closed curves characterizes the fact that any two curves in the arrangement have at most two crossings. Our studies provide the necessary ideas for exhibiting a drawing of K_10 that has no extension to an arrangement of pseudocircles and a drawing of K_9 that does extend to an arrangement of pseudocircles, but no such extension has all pairs of pseudocircles crossing twice.

## Authors

• 5 publications
• 1 publication
• 1 publication
09/16/2019

### Extending simple drawings with one edge is hard

A simple drawing D(G) of a graph G = (V,E) is a drawing in which two edg...
03/11/2022

### Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs

Simple drawings are drawings of graphs in which the edges are Jordan arc...
12/06/2017

### Arrangements of Pseudocircles: On Circularizability

An arrangement of pseudocircles is a collection of simple closed curves ...
07/17/2019

### An Ongoing Project to Improve the Rectilinear and the Pseudolinear Crossing Constants

A drawing of a graph in the plane is pseudolinear if the edges of the d...
02/28/2022

### Parameterised Partially-Predrawn Crossing Number

Inspired by the increasingly popular research on extending partial graph...
12/11/2018

### Bundled Crossings Revisited

An effective way to reduce clutter in a graph drawing that has (many) cr...
08/01/2019

### On the existence of paradoxical motions of generically rigid graphs on the sphere

We interpret realizations of a graph on the sphere up to rotations as el...
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## 1 Introduction

The Harary-Hill Conjecture states that the crossing number of the complete graph is given by the formula

 H(n):=14⌊(n(2⌋⌊n−12⌋⌊n−22⌋⌊n−32⌋.

Some of the known families of drawings of achieving crossings have the geometric character of being spherical: a spherical drawing of a graph is a drawing in the unit sphere in which the vertices of are represented as points—no three on a great circle—and the edges of are shortest-arcs in connecting pairs of vertices.

Examples of such families are: Hill’s Tin Can Drawings [15]; Kynčl’s general spherical drawing in his posting [17]; and, most recently, the family of Ábrego et al. [3] in which every edge is crossed at least once. Even though there are spherical drawings of with crossings, the Harary-Hill Conjecture remains unknown for this special class of drawings. It is not clear to us whether a 2-page-drawing of having crossings is necessarily spherical. See [4].

A surprising result by Moon [20] states that the number of crossings in a random spherical drawing of has, as goes to infinity, crossings. Thus, spherical drawings are linked with the asymptotic version of the conjecture:

 limn→∞cr(Kn)(n4)=38.

A rectilinear drawing of a graph is a drawing of in the plane so that its edges are straight-line segments. One of the most important recent accomplishments in the study of crossing numbers is a result of Lovász et al. [18] and, simultaneously and independently, Ábrego and Fernández-Merchant [1], showing that rectilinear drawings have at least crossings. (It is now known that, for , rectilinear drawings have strictly more than crossings.)

The proofs that rectilinear drawings of have at least of crossings use machinery for studying arrangements of pseudolines, based on the property that the edges in a rectilinear drawing can be extended to such an arrangement. An analogous property is satisfied by spherical drawings: each edge can be extended to a great circle (a circle in of maximum diameter, and the edge is the shorter side joining the two vertices). This means that the edges can be extended into an arrangement of pseudocircles, defined as a set of simple closed curves in such that every two intersect at most twice, and every intersection is a crossing between two curves.

The following questions are principal motivations for this work.

1. [label=Question 0., ref=,leftmargin=75pt,topsep=-0pt,itemsep=0pt]

2. Is every optimal drawing of spherical?

3. Is the spherical crossing number of equal to ?

The success of the geometric approach for the rectilinear crossing number of suggests trying an analogous approach for spherical drawings, replacing arrangements of pseudolines with arrangements of pseudocircles. A principal goal of this work is to introduce pseudospherical drawings as a generalization of spherical drawings.

There are nice characterizations of pseudolinear drawings of . Aichholzer et al. [5] prove that a drawing of in the plane is pseudolinear if and only if every crossing has the facial 4-cycle bounding the infinite face of the . Arroyo et al. [7] have an equivalent characterization based in their notion of a convex drawing of . Arroyo et al. [6] characterize when a drawing of a general graph in the plane is pseudolinear.

Therefore, the other principal goal of this work is to characterize pseudospherical drawings of ; this is also based in the notion of convex drawings mentioned in the preceding paragraph.

Motivated by the remarks above, a drawing of a graph in the sphere is pseudospherical if, for any distinct edges and :

1. [leftmargin=60pt,label=(PS0), ref=(PS0),itemsep=0pt]

2. is contained in a simple closed curve such that no vertex other than an end of is contained in ;

3. and all intersections are crossings; and

4. has at most one point.

A drawing is weakly pseudospherical if it satisfies 1, 3, while 2 is relaxed to

1. [leftmargin=60pt,label=(PS0w), ref=(PS0w),itemsep=0pt,start=2]

2. and all intersections are crossings.

Our main result is the following characterizations of pseudospherical drawings of . (The definition of h-convex will be given in Section 4.)

###### Theorem 1.1.

For a drawing of , the following are equivalent:

1. [leftmargin=60pt, label=(1.1.0),ref=(1.1.0)]

2. is pseudospherical;

3. is weakly pseudospherical; and

4. is h-convex.

It is not difficult to prove that the Blazěk and Koman 2-page drawing of having crossings is weakly pseudospherical [4, 11]. Theorem 1.1 shows that it is, therefore, pseudospherical and h-convex.

The implication 12 is trivial. Although the reader cannot see it now, the implication 23 is quite easy. The hard part is 31. We do not see how to prove 21 directly.

The proof of 31 proceeds by iteratively finding a pseudocircle for the next edge to extend by one the set of pseudocircles satisfying the conditions 13. There are two principal steps involved. The first step is to find two initial approximations for the next pseudocircle (when added to , either of these will satisfy 1), while the second is to repeatedly shift one of the initial approximations, gradually increasing the number of pseudocircles in that it intersects until it intersects them all, at which point it is a -transversal.

Each of these steps has its challenges. To find the initial approximations, we require an extensive study of h-convex drawings; this is done in Section 4. Crucially, each edge of an h-convex drawing partitions the vertex set of into two pseudolinear drawings of (typically smaller) complete subgraphs; the initial approximations are near the outer boundary of each of these pseudolinear subdrawings. (Motivated by this study, we tried to show this partitioning holds for pseudospherical drawings of general graphs. In a personal communication, Xinyu Lily Wang has shown that it is false in the more general context.)

Producing the pseudocircle transversal requires shifting one of a pair of initial approximations from the preceding paragraph towards the other using analogues of Reidemeister II and III moves. The core of the shifting turns out to require a characterization of arrangements of pseudocircles.

Let be an arrangement of simple closed curves in the sphere; that is, a set of simple closed curves, any two of which have finitely many intersections, all of which are crossings. A spiral of is an arc in the union of the curves in that always makes the same—either all left or all right—turn in changing from one curve to another (see Figure 1). In Section 2, we give a more precise definition of spiral and define coherent spirals. The auxiliary result that we need is the following.

###### Theorem 1.2.

Let be an arrangement of simple closed curves. Then is an arrangement of pseudocircles if and only if has no coherent spirals.

Our study of spirals led us to two drawings, one for each of and . The former has no extension of its edges to an arrangement of pseudocircles. The latter has such an extension, but no extension has all pseudocircles crossing exactly twice. These examples are exhibited in Section 7.

Independently, Aichholzer at the 2015 Crossing Number Workshop in Rio de Janeiro and Pilz at the Geometric Graph Week in Berlin (2015) asked if every drawing of has an extension to an arrangement of pseudocircles. The drawing of answers this question in the negative. The drawing of answers negatively the related question of improving a pseudocircular extension to a pseudospherical extension.

In the next section, we introduce spirals and prove Theorem 1.2. Section 3 contains the proof that an initial pair of approximations for implies the existence of a -transversal. Section 4 introduces h-convex drawings and proves the easy implication 23. Section 5 contains the necessary discussion of h-convex drawings for the initial approximations. The proof of Theorem 1.1 is completed in Section 6 by showing how to inductively obtain in an h-convex drawing the initial curves for the sweeping. The interesting drawings of and are in Section 7, while Section 8 has concluding remarks.

## 2 Spirals and Coherence

A spiral is a special arc, illustrated in Figure 1 and defined precisely below, in the union of an arrangement of simple closed curves that has all its crossings facing the same side of the arc. (Alternatively, a spiral always makes the same—either all left or all right—turn in changing from one curve to another.) The main result in this section is a characterization of arrangements of pseudocircles as either (a) not having any “coherent” spirals or, equivalently, (b) every spiral has an “external segment”. That an arrangement of pseudocircles has no coherent spiral is the point required for the proof of Theorem 1.1.

In an arrangement of simple closed curves (defined just before Theorem 1.2), three or more of the curves in may cross at the same point. This is necessary for extensions of drawings of , where curves pairwise cross at each vertex.

A non-trivial arc in the union of the curves in with ends and has a unique decomposition sequence of subarcs of , as depicted in Figure 2, such that:

1. [label=(),leftmargin = .50 truein,ref=()]

2. is an end of , is an end of ;

3. for each , there is a such that ; and

4. for , the curves and in are distinct and is a crossing of and .

The number is the weight of . Figure 2 shows an arrangement of simple closed curves with an arc of weight 3. For , the crossing of with is denoted . For convenience, we set and .

For , and are the closures of the components of incident with and , respectively. Four such are illustrated in Figure 3. Evidently, consists of the continuation of through either () or () up to the next meeting with .

The continuations and both leave on the same side of . This is the side of that faces. In Figure 2, and face different sides of the dotted arc. We are now prepared for the definitions of spiral, external segment, and coherence.

###### Definition 2.1 (Spiral, External Segment, Coherence).

Let be an arrangement of simple closed curves and let be an arc in with decomposition sequence . For each , let be such that . (Only consecutive are required to be distinct; if , then could be the same as .)

1. [label=(2.1.0),leftmargin = 60pt,ref=(2.1.0)]

2. The arc is a spiral if all of face the same side of .

3. For , the segment is external for if ; otherwise, is internal for .

4. For every and , the arc is a coherent extension if: (a) is internal for and (b) has both ends on the same side (see discussion of “side” below) of the interior of . (In Figure 3, is not a coherent extension; the other three are.)

5. is coherent if at least one of and is a coherent extension. In Figure 3, both and are coherent.

6. is coherent if, for each , is coherent.

As we traverse from one end to the other, there are naturally left and right sides. The two ends of the arc are in . The issue in the definition of coherence is: are the points near each end of on the same side of or not. “Left” and “right” depend on an orientation of and are irrelevant to us.

To be on the same side, the ends of must be in the interior of . As the points and are not in the interior of , we have the following.

###### Remark 2.2.

Neither nor is coherent.

The following characterization of an arrangement of pseudocircles in terms of its spirals seems to be quite interesting in its own right. We only need 13 for the proof of Theorem 1.1.

###### Theorem 2.3.

Let be an arrangement of simple closed curves in the sphere. Then the following are equivalent:

1. [label=(2.3.0), ref=(2.3.0), leftmargin=60pt]

2. is an arrangement of pseudocircles;

3. has no coherent spirals with weight 1;

4. has no coherent spirals; and

5. every spiral in has a segment external for .

This characterization fits in well with much recent work on arrangements of pseudocircles. The paper of Felsner and Scheucher [12] is one example; their references [2], [7], [10], [15], [19], and [26] are others. Felsner and Scheucher have a web page devoted to pseudocircles [13]. Another recent work about unavoidable configurations in the sense of Ramsey’s Theorem is Medina et al. [19].

The example to the left in Figure 4 has a spiral in an arrangement of pseudocircles. This spiral is not coherent because is external for .

In addition to its use in the proof of Theorem 1.1, Theorem 2.3 is useful for constructing drawings of graphs that cannot be extended to arrangements of pseudocircles. For example, the right-hand diagram in Figure 5 cannot be extended to an arrangement of pseudocircles because the -arc indicated by the dashed curve induces a coherent spiral in any such extension. A similar idea is used in Section 7 to construct a non-extendible drawing of .

Proof of Theorem 2.3.  The implications 43 and 32 are trivial. We prove 21 and 13 to complete the proof that 1, 2, and 3 are equivalent and finish the proof with 14.

21. Suppose by way of contradiction that are distinct curves in such that . Then . Let and, traversing in one direction starting at , let be the first three points of encountered.

Let be the arc obtained by starting at , continuing along through to and then following through to a point just beyond . The decomposition of is , with, for , . Since every arc in with weight at most 1 is a spiral, is a spiral. The fact that the points and of are consecutive in imply that is coherent.

On the other hand, the -subarc of intersects the -subarc of just in . This shows that is coherent, completing the proof that is coherent, the required contradiction.

13. To obtain a contradiction, suppose is a coherent spiral with least weight. An arc with weight 0 is not coherent, so the decomposition of has .

The first claim imposes constraints on what happens at points “under” a forward jump such as, in Figure 3, under . For an extension of , denotes the other end of .

###### Claim 1.

Suppose that is a coherent extension of . If there is an such that is in , then:

1. [ref=(0)]

2. for each , is disjoint from ; and

3. either: for some , ; or intersects .

Likewise, suppose is a coherent extension of . If there is an such that is in , then:

1. [start=3,ref=(0)]

2. for each , is disjoint from ; and

3. either: for some , ; or intersects .

Proof. We prove the first statement; the “likewise” is the same, but for the traversal of in the reverse direction.

By way of contradiction, suppose first that, for some , intersects ; let be the first intersection with as we traverse from . The interior of the subarc of from to is on the side of the unique simple closed curve contained in that is opposite to the side that contains .

See Figure 5 for an illustration of this proof.

Let be the subarc of consisting of and the portion of from to . Likewise, let be the subarc of consisting of and . Then the arc consisting of segments has smaller weight than . Also, even if , is a spiral. (In case , then , , and all cross at . This ensures that the cyclic rotation of these three curves at is , where is the continuation of from .)

To see that is coherent, first let be the simple closed curve . For each of the segments of , contains some segment of . Let be a coherent extension of for . Follow from its end in (or, if , from ). If we never encounter , then we arrive at on the same side. On the other hand, if we encounter , it is not at a point in and so it is in .

Label as the outside of the side of containing . Since starts on the outside of , its first intersection with is from that side. Thus, the portion of up to that first intersection with is a coherent extension of , as required.

To complete the proof that is coherent, we note that, if, for the segment of , both and are coherent, then they are contained in coherent extensions of the corresponding segment of . Since these extensions for are distinct, as extensions for they are also distinct. Thus, there is a coherent spiral with weight less than , a contradiction.

For 2, if, for each , and is disjoint from , then and (using 1) each is a coherent extension in . The same argument as in the preceding paragraph shows that is a coherent spiral with smaller weight than , a contradiction.

The next claim considers “reverse” coherent extensions. This claim rules out the possibility of an extension such as in Figure 3.

###### Claim 2.

There do not exist such that and is a coherent extension with an end in .

Likewise, there do not exist such that and is a coherent extension with an end in .

Proof. We only prove the first statement. Choose the least for which such an exists. Suppose first that . Thus, . Because , . Thus, Hypothesis 1 implies . Let be the simple closed curve contained in . Except for , is disjoint from .

Note that is not in . Therefore, and the start of from are on different sides of , which is impossible. Thus, .

The choice of implies that either is not coherent or it does not intersect . Therefore, must intersect at a point other than . This gives the two crossings of with .

An intersection of with yields a third intersection of with , and the theorem is proved. Therefore, we assume is disjoint from . On the other hand, the choice of implies that, for each with in the interior of , . Therefore 2 of the preceding claim shows is not disjoint from , the final contradiction.

We may now suppose that there does not exist an that is a coherent extension with an end in any such that . Similarly, we may assume that there does not exist an that is a coherent extension with any end in any such that .

The final claim combines the first two to completely determine the nature of a coherent extension. Before we get to it, we require one more detail.

###### Claim 3.

Let . Suppose is not a coherent extension and that has an intersection with . Then is not coherent.

Likewise, if is not coherent and has an intersection with , then is not coherent.

Proof. We only prove the first statement. Because is coherent and is not a coherent extension, is a coherent extension of . Let be the intersection of with ; it follows that . Since , is neither nor . Hypothesis 1 implies .

Since is disjoint from ; in particular, it is disjoint from . For , the preceding claim implies that is not coherent. Thus, we suppose .

The union of and is a simple closed curve . Let be a point of near . From trace an arc alongside , across and, continuing beside , on to a point near the end of . Thus, is along the side of faced by all the . Because is not coherent, it does not return to on this side and, therefore, crosses only once. Consequently, crosses only once.

Suppose by way of contradiction that is a coherent extension of . Because does not intersect , it cannot cross . Therefore, it does not intersect the portion of from its crossing with to its end near . In particular, has no end in . The first paragraph shows is also disjoint from . Thus, is in , contradicting the preceding claim.

We are now ready to get the fine detail of the coherent extensions of .

###### Claim 4.

For , if is a coherent extension of , then .

Likewise, for , if is a coherent extension of , then .

Proof. We only prove the first statement. Suppose that is least such that is a coherent extension of and . We show by induction that, for each , is not a coherent extension of . For , Remark 2.2 shows that is not a coherent extension of . Now let and suppose that is not a coherent extension of .

Since is coherent and is not a coherent extension of , we have that is a coherent extension of . By the choice of , . The preceding claim implies that is not a coherent extension of , completing the proof that, for each , is not a coherent extension of .

Let be such that . The second claim and the choice of show that . Part 1 of the first claim implies that, for each , is disjoint from , while the second claim asserts that . Part 2 of the first claim now implies that intersects at a point .

We showed that is not a coherent extension of and that intersects at and . Consequently, the coherence of and the preceding claim show that is a coherent extension. The second claim shows that, for some , . The first two claims show that, for each , is disjoint from .

Let be the subarc and let be the arc consisting of , . Just above, we showed that is disjoint from . Therefore, is a coherent extension of with respect to . The second claim shows that has both ends in .

Since intersects in and , we see that ; therefore . Moreover, it follows that is disjoint from . This, together with the fact that, for each , is disjoint from , contradicts the first claim.

Because is coherent and Remark 2.2 shows is not a coherent extension of , is a coherent extension of . Likewise, is a coherent extension of . It follows that there is a such that is a coherent extension of and is a coherent extension of . The fourth claim implies that both ends of are in and both ends of are in . This implies that , completing the proof that 13.

14. For this argument, we will make use of the following trivial lemma.

###### Lemma 2.4.

Let be arcs in the sphere such that and both have their ends in , but otherwise are disjoint from . We assume and have finitely many intersections and these are all crossings.

Assume that short subarcs of starting at each end of are on different sides of the unique simple closed curve in . Then has at least one point.

By way of contradiction, let be a least-weight spiral in having no segment external for . Since the only segment in an arc of weight 0 is external for that arc, has positive weight . Let be the decomposition of .

If any spiral in is coherent, then 13 implies that some two curves in have at least four crossings, a contradiction. In particular, we may assume is incoherent; let be an incoherent segment of . By definition, both and are both incoherent extensions. Thus, short subarcs near their ends and are on the side of that is opposite the side faced by all the crossings . We remark that one or both of and might be in . If , then only the subarc near is forced by incoherence to be on the side of not faced by the crossings; the fact that is a simple closed curve implies is as well. An analogous statement applies if .

As we traverse from to , we first encounter and then (these could be equal). Therefore, either or . As these are symmetric up to reversal of , we assume the latter.

Let be the spiral ; evidently, its weight is less than that of . Therefore, has a segment that is external for . Notice that the side of faced by all its crossings is separated from by the simple closed curve .

Since no segment of is external for , intersects at a point outside . This implies that and intersect at points and , respectively, that are not in . Since is external for , and are in .

###### Claim 5.

Traversing and from and , respectively, they each have an intersection with and these intersections are distinct points of .

Proof. In an extreme case, ; here is the required common point between and . In the other extreme case, and ; here is the common point between and . (These could happen simultaneously.) Otherwise, and are interior points of .

If we take , and to be either or , then our trivial lemma implies there are distinct points, one in each of and .

Since , the only points in are the points in each of and . In particular, and are both disjoint from .

Because and is disjoint from , . On the other hand, the disjointness of with implies that is not further from in than is. In turn, this implies that is in . Therefore, reversing the direction of traversal of , may play the role of in the preceding argument.

Thus, there is a such that is a segment of , external for , but not external for . The argument above for and applies to and . Thus, and have their endpoints and , respectively, in . The trivial lemma implies that each of and has an intersection with each of and . In particular, has at least four points, as required.

## 3 A pseudocircle transversal

We recall that the proof of Theorem 1.1 has two parts. For our current set of pairwise intersecting pseudocircles, we must (i) find a pair of initial approximations to the pseudocircle for the next edge and (ii) show that the pair of approximations imply the existence of the desired curve crossing all the curves in . In this section, our focus is on the second of these.

In particular, in Section 6, we will show how to find two initial approximations that together intersect all the curves in our current . The main result of this section is to use these two approximations to find the single curve that intersects all the curves in .

We are reminded of the theorems that are:

• [topsep=0pt, itemsep=-4pt,leftmargin=11pt]

• Helly-type: if a collection of sets is such that every of the sets admits a transversal, then the whole collection admits a transversal; and

• Gallai-type: if a collection of sets is such that every of the sets admits a transversal, then the whole admits a small set of partial transversals whose union is a transversal.

Our theorem has the following different character: if a collection of sets admits a small set of partial transversals whose union is a transversal, then it admits a transversal. We do not know of an another example of this type of theorem.

###### Definition 3.1.

Let be an arrangement of pseudocircles.

• A set of simple closed curves is a -transversal if every curve in intersects at least one of the curves in .

• A simple closed curve is a -pseudocircle if is an arrangement of pseudocircles. [Text removed.]

###### Theorem 3.2.

Let be an arrangement of pseudocircles. Let and -pseudocircles such that is a -transversal. Suppose

1. [label=(3.2.0),leftmargin=63pt,ref=(3.2.0)]

2. is a non-trivial arc and

3. if are such that and , then .

Then there exists a -pseudocircle containing and is contained in the closure of the face of not incident with .

Our proof shows that one can sweep either of or to the required . In their classic paper [22], Snoeyink and Hershberger show how to sweep one curve through the others in an arrangement of pseudoarcs and pseudocircles. In particular, up to sweeping, is unique.

To prove Theorem 1.1, the current is complete, so Hypothesis 2 holds automatically.

Proof.  Let and, for , let . Because is a -transversal, . Let . Define as the number of crossings in included in the face of not incident with . We proceed by induction on .

We can assume that neither nor is empty, else we pick to be either equal to or .

If there is an arc of some incident with a face of that is included in and incident with , then by shifting some part of to cross via a Reidemeister Type II move, we obtain a curve such that the pair satisfies the same hypothesis as . Since , the result follows by induction.

In the alternative, there exists an arc with ends in , but otherwise contained in , such that separates from . Let be the closure of the component of that is incident with both and . Among the finitely many choices for , we choose so that is minimal under inclusion. See Figure 6.

We apply Theorem 2.3, specifically 13, to see that there is a crossing in facing . The proof is by contradiction, assuming that every crossing faces the other side of . In particular, is a spiral; the contradiction follows from the following.

###### Claim 1.

There is a crossing in facing .

Proof. By way of contradiction, suppose every crossing faces the other side of ; in particular, is a spiral. We show that is coherent, contradicting Theorem 2.3 (specifically, 13). It suffices to show that if is an arc in the decomposition of , then is coherent. Let be the curve in containing .

Let be the closed arc in such that is a simple closed curve. Consider the continuation of from one end of . Since crosses twice, the continuation must eventually reach ; in particular, it must have a first intersection with .

We show below that it is impossible for both continuations to have these first intersections in . Therefore, for one of them, the first intersection is in the interior of , showing that is coherent.

So suppose both continuations intersect for the first time at and . Since , . Therefore, is contained in the region bounded by and intersects the boundary of this region only at . This shows that , the required contradiction.

###### Claim 2.

Let . Then every arc in has one end in the interior of and one end not in the interior of .

Proof. Let be an arc . If has no end in , then contains an arc that separates from such that is properly contained in , contradicting the choice of .

If has no end in the complement of the interior of in the boundary of , then is contained in and has both ends in the interior of . Since has exactly two points, these are the two ends of . The preceding paragraph shows that is the only arc in , and hence is included in the side of disjoint from . Thus, for any


text:
# Annotating LC/MS data with cliqueMS In cliqueMS: Annotation of Isotopes, Adducts and Fragmentation Adducts for in-Source LC/MS Metabolomics Data

knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )  # Introduction Untargeted metabolomics goal is to quantify as metabolites as possible from a sample. We can use liquid chromatography coupled to mass spectrometry (LC/MS) for this purpose. It is a great challenge to transform LC/MS data into a profile of annotated metabolites that provides us meaningful biological information. A very important limitation to the annotation of metabolomic experiments is that the number of m/z processed signals, called features, is much bigger than the putative number of metabolites in a sample. The sources that produce multiple features from a single metabolite are multiple and variable. Natural isotopes as carbon isotopes produce isotope features. Ionization of metabolites produce the so called adducts of the metabolite, which are detected as different features depending on the ion adduct involved ([M+Na]+, [M+H]+, etc..). Apart from adduct features, ionization also produces metabolite fragmentation and other reactions as dimerizations, trimerizations, all of them being detected as multiple features. Being the reduction of multiple features to single metabolites a crucial step for the correct annotation of LC/MS experiments, we will show how to use cliqueMS to do so. # Annotating features in LC/MS metabolomics cliqueMS annotates samples one by one. Annotation can be summarised in these three steps: 1. Divide features into clique groups. 2. Annotate isotopes. 3. Annotate adducts and fragments Annotation steps are stored in an anClique S3 object. This object can be created from a XCMSnExp or a xcmsSet object with processed m/z data from xcms package. First m/z raw data is processed: library(cliqueMS) mzfile <- system.file("standards.mzXML", package = "cliqueMS") library(xcms) mzraw <- readMSData(files = mzfile, mode = "onDisk") cpw <- CentWaveParam(ppm = 15, peakwidth = c(5,20), snthresh = 10) mzData <- findChromPeaks(object = mzraw, param = cpw)  Then we can create an anClique object: ex.anClique <- createanClique(mzData) summary(ex.anClique)  Here we see an anClique object before any annotation step. Features have not been grouped, isotopes and adducts are not annotated. Now let's see the three steps in detail. ## Grouping features Metabolites produce multiple features and very often they do not separate completely in the chromatography, so we observe coelution. This increases the difficulty of the annotation because many features coming from different metabolites might appear very close in the chromatogram. Before trying to annotate isotopes, adducts and fragments we want to make groups of features. Ideally, each group should include all the features produced by a single metabolite. ### A network based algorithm to find groups of features cliqueMS uses a similarity network to find groups of features. Each feature is a node, and edges are weighted according to the cosine similarity between features:$c_{ij}=\frac{\sum_k f_i(t_k)f_j(t_k)}{\| {f}_i\| \|{f}_j\|}$Values from cosine similarity are useful to discriminate pairs of features that come from the same metabolite from pairs of features that come from different metabolites^[1]. We compute the cosine similarity using the profile mode of the data, having each feature a m/z value and vector intensities. All features are discretized into a vector of equal bins$k$. Each vector position relative to retention time$t_k$contains the intensity of the feature$i_k$at that moment of the chromatography. Features with no coelution at all have a cosine similarity = 0. Edges with weight = 0 are not included in the network, nor nodes without any edge. Once we have the network, it is time to divide the features into groups. cliqueMS assumes that the similarity between all features that come from the same metabolite must be$c_{ij} > 0$. Additionally, similarity values between features of the same metabolite should be generally higher than between features of different metabolites. With this information, cliqueMS uses a probabilistic model to find the feature groups. This model find cliques, fully connected components so for all nodes$c_{ij} > 0$. The similarity inside this cliques should be higher than the similarity between features outside the clique. cliqueMS estimates the log-likelihood for a particular assignment of features into different clique groups. For details in the probabilistic model and the log-likelihood maximisation see [1]. The log-likelihood maximisation procedure can be summarised in the following way: 1. cliqueMS starts with each node as a different clique group. 2. Alternate between merge cliques and move nodes from cliques if the new assignment has a bigger log-likelihood. 3. When log-likelihood cannot be increased in 2, try to move each node from its clique to another, ("Kernighan-Lin"). 4. Final group assignment for all features. Now let's see how to find this groups with getCliques. ### getCliques With the function getCliques we assign clique groups to our features. This function creates the network of similarity and then computes the clique groups. As input data it uses a xcmsSet object. getCliques outputs an anClique S3 object, which will be used to store all annotation steps. set.seed(2) ex.cliqueGroups <- getCliques(mzData, filter = TRUE) summary(ex.cliqueGroups)  As we see from the printed messages, the function getCliques first creates a network, and then it filters features if parameter filter = TRUE. As m/z signal processing algorithms may produce two artefact features from a single one, it is recommended to set filter = TRUE to drop repeated features. This filter only drops features with similarity > 0.99, and equal values of m/z, retention time and maximum intensity, defined by the relative error parameters mzerror, rtdiff and intdiff. From the output of the function we see the computed log-likelihood at the beginning, when each feature is in a different clique and the computed log-likelihood at the end of the process. If we now look at the summary of the resulting anClique object, we see that the features have been grouped into 69 clique groups. Now we can annotate isotopes. ## Annotating isotopes cliqueMS annotates isotopes within each clique group. cliqueMS searches pairs of features than can be carbon isotopes based in these two rules: 1. The monoisotopic feature must be more intense than the isotopic feature. 2. The mass difference between the features must be the difference of a carbon isotope$\pm$an error. Isotopes are annotated with the function getIsotopes. This function finds pairs of features that fulfil the conditions of an isotope. Then it creates the isotope annotation after removing incoherences like two monoisotopic masses for one isotope, two second isotopes for one first isotope, etc... In all this cases the removed pair is the one with smaller similarity. The use of the function is pretty straightforward: ex.Isotopes <- getIsotopes(ex.cliqueGroups, ppm = 10) summary(ex.Isotopes)  Parameter ppm is important because it defines in ppm units the range of the accepted relative error. Once isotopes are annotated we can annotate adducts and fragments. ## Annotating adducts and fragments ### Putative adducts The last step of ´cliqueMS´ is to annotate adducts. Each feature has a m/z value that is the neutral mass of the metabolite plus the mass of the ion adduct (or fragmentation ion adduct). The neutral mass is an unknown value, but the ion adduct mass is to some degree known as many ion adducts are known. The list of possible adducts should be given as input to cliqueMS by the user or either use one of the default adduct lists. Here is how it looks the positive default list: head(positive.adinfo)  Which contains the name of the adduct, the log10 frequency of that adduct, the mass of the adduct, the number of molecules involved and the charge (see [1] for details in how default lists were made). With the adduct list we can obtain neutral masses. ### Scoring neutral masses cliqueMS searches in each clique for groups of two or more features compatible with a neutral mass and two or more adducts in the adduct list. Neutral masses with only one adduct are not included in the scoring. Once we have all possible neutral masses and their corresponding adducts, the algorithm tries combinations of different adducts and neutral masses to find the most plausible annotation. All combinations are scored and the top five annotations are returned. The scoring is based on the following criteria: 1. The log-frequency of the adduct 2. Minimum number of empty features 3. Minimum number of neutral masses The computed score (which is a logarithmic score) is the sum of the adducts log-frequencies plus the number of empty features (which has a log-frequency smaller than the least frequent adduct) and the number of neutral masses. Within a clique group, it may happen than the annotation of some features is independent from the annotation of some other features, as there is not a single neutral mass with adducts in both groups of features. In those cases, the clique group is splitted in non overlapping groups, called annotation groups. This is common in big cliques. The reported scores refer to annotation groups. The score is useful to see how probable is the first annotation compared to second annotation, third annotation, etc... within an annotation group, but it is not intended to compare annotations between different annotation groups because the score will be smaller when the number of features in the group is bigger. To find annotation cliqueMS uses the function getAnnotation. ### getAnnotation Here is an example of annotating adducts with getAnnotation ex.Adducts <- getAnnotation(ex.Isotopes, ppm = 10, adinfo = positive.adinfo, polarity = "positive") summary(ex.Adducts)  As we see from the summary output, 178 of 275 features have annotation. Function getAnnotation requires as input an adduct list, the parameter adinfo. Users can use the default adduct list positive.adinfo for positive charged adducts and and negative.adinfo for negative charged adducts. polarity must be set, either to positive or negative. Lots of neutral masses are found when the clique groups have many features. In those cases, scoring all annotations could take much time as there are many possible combinations. To prevent this, neutral masses that likely will be in the final top annotations are selected and annotation is computed quickly. The selected masses have the highest frequency adducts and the largest number of adducts. For each clique group, all neutral masses are ordered depending on their score. A number of top scoring masses controlled by topmasstotal parameter are selected. Additionally and for every feature, the ordered list of scored neutral masses is subsetted to only the neutral masses with adducts in that feature. Then a number of top scoring masses set by topmassf parameter are selected in each sublist, and added to the set of selected masses. After the mass selection, and in cases of big cliques (size of a "big" clique is defined by parameter sizeanG), annotation groups are splitted again in new non overlapping groups just taking into account the set of selected neutral masses. getCliques stores the annotation in the peaklist of the anClique object. Here we can see an overview of some annotated features in our sample: features.clique6 <- ex.Adducts$cliques[[6]]
c("an1","mass1","an2", "mass2", "an3", "mass3", "an4", "mass4", "an5", "mass5")], n = 10)

Now we have obtained the neutral mass and the adduct annotation for our features. We could use the neutral mass together with the retention time and MS/MS data to annotate more confidently some of these metabolites. We also know how many features in the dataset are isotopes. Finally, we have achieved a reduction in the complexity of our the dataset, from many features to a signficant smaller number annotated neutral masses that have different adducts and isotopes.

1 - "AddClique, a network-based algorithm for the identification of adducts in LC/MS metabolomics", Senan O, Yanes O, Sales-Pardo M, Guimerà R. et al, Metabolomics Congress 2016 Dublin.

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cliqueMS documentation built on May 1, 2019, 6:32 p.m.