text:
Recommendations for Management of Whiteflies, Whitefly-Transmitted Viruses, and Insecticide Resistance for Production of Cucurbit Crops in Florida
Xavier Martini, Mathews Paret, Jane E. Polston, Scott Adkins, Pamela Roberts, Teresia Nyoike, and Oscar E. Liburd
Historically, squash was the only cucurbit crop seriously affected by the Middle East Asia Minor 1 (MEAM 1) sweetpotato whitefly, Bemisia tabaci (Gennadius) (formerly known as the silverleaf whitefly, B. argentifolii and Bemisia tabaci Biotype B), because of the silverleaf disorder induced by feeding of the immature stages (nymphs). However, since 2004 four viruses transmitted by the whitefly were found in watermelon, melon, and squash in Florida, making whitefly management a priority for most cucurbits.
Squash vein yellowing virus (SqVYV), an ipomovirus in the family Potyviridae (the same family that includes many of our aphid-transmitted viruses), was first identified from squash in Florida in 2005 but had been causing serious damage in watermelon as watermelon vine decline since at least 2003 (Adkins et al. 2007). Unlike aphids, which can acquire Zucchini yellow mosaic virus in a matter of seconds just by tasting the plant, whiteflies must feed on the plant to acquire SqVYV, and the longer they feed (up to about 8 hours), the more efficiently they are able to transmit the virus to a new plant. Whiteflies appear to retain the virus for less than a day after they move away from an infected plant, but SqVYV can still spread rapidly when whitefly populations are high. Watermelon plants are killed by the virus (Figure 1), and fruit often have necrotic areas (Figure 2), especially just under the rind. Early symptoms on watermelon include yellowing, curling of the new growth, and collapse of petioles (Figure 3). Squash plants develop yellowing along the leaf veins (Figure 4) and are stunted if infected early. Symptoms are more severe on yellow squash, including green streaks on the fruit, than on zucchini. Butternut squash, pumpkin, and Calabaza, all Cucurbita species, are also affected. Cucumber and muskmelon (Cucumis species) are not very susceptible to the virus. So far, the only weed hosts known are restricted to the cucurbit family: balsam apple, creeping cucumber, and smellmelon (Adkins et al. 2008; 2009).
Figure 1. Watermelon plants killed by infection with SqVYV.
Credit: Pamela Roberts UF/IFAS
Figure 2. Effects of infection with SqVYV on watermelon fruit. Note necrotic (brown) areas of rind.
Figure 3. Early symptoms of infection with SqVYV in watermelon.
Figure 4. Vein yellowing symptoms caused by SqVYV on yellow summer squash in the greenhouse.
Cucurbit leaf crumple virus (CuLCrV) was first recognized in 1998 in the Imperial Valley of California. CuLCrV was first found in northeast and north-central Florida in 2006 but has now spread to other parts of the state. It is a begomovirus in the family Geminiviridae. Once a whitefly acquires CuLCrV, it can transmit the virus for the rest of its life. CuLCrV causes a suite of symptoms primarily characterized by leaf chlorosis, distortion, curling and crumpling, and plant stunting and wilting, leading to significant reduction in total yield. It appears to be more damaging to squash than to cantaloupe and watermelon; however, internal fruit quality issues (discoloration and reduced sugar) have been recently found on CuLCrV-infected watermelon. The fruit of yellow squash plants develop green streaks. In watermelon, yellowing of leaves and crumpling can occur (Figure 5). All cucurbits are susceptible except for acorn and butternut squash and a number of melons, such as galia, honeydew, casaba, and golden crenshaw. CuLCrV has also been found to infect green beans, balsam apple, and smellmelon in Florida (Akad et al. 2008; Adkins et al. 2008, 2009). Other weed hosts have not yet been identified. See https://edis.ifas.ufl.edu/publication/in716 for other photographs of symptoms in squash, watermelon, and snap bean.
Figure 5 . CuLCrV infection symptoms in watermelon.
Credit: Xavier Martini, UF/IFAS
Cucurbit yellow stunting disorder virus (CYSDV) was originally reported from the Middle East and Spain. It was first found in the United States in Texas in 1999 and was widespread in the southwestern United States in the fall of 2006 (Brown et al. 2007; Kuo et al. 2007). CYSDV was found in Florida in 2007 (Polston et al. 2008). It is a crinivirus in the family Closteroviridae. CYSDV can persist in the whitefly adult for up to 9 days, the longest time of any crinivirus (Wisler and Duffus 2001). It affects all cucurbit crops, causing reduced fruit size and sugar content. Symptoms resemble those of a nutritional deficiency or water stress; the most obvious symptoms are seen on older leaves. The first symptom is a yellow spotting of leaves (Figure 6). Then, leaves turn yellow between the leaf veins, which remain green—the opposite of what happens when squash is infected with SqVYV. Leaves roll downward and become brittle. It can take up to four weeks for symptoms to develop after infection, so it is easy to move the virus long distances in infected but symptomless transplants without realizing that the plants are infected. CYSDV has also been found infecting smellmelon in Florida (Adkins et al. 2009).
Figure 6. CYSDV symptoms in watermelon.
Credit: Pamela Roberts, UF/IFAS
Cucurbit Chlorotic Yellows Virus (CCYV) is the more recent arrival in Florida, as it was first discovered in 2021 in north Florida and identified also in watermelon grown in south Florida (Jailani et al. 2021; Hendricks et al. 2021). It is a Crinivirus in the family Closteroviridae. Symptoms observed include leaf crumpling, yellowing and curling, and vein yellowing. These symptoms are similar to those found in single/and or mixed infection of CuLCrV, CYSDV, and SqVYV. The first outbreak observed had up to 100% disease incidence in the field that led to fruit quality issues and yield losses of 80% and above.
Figure 7 . (A) CCYV damage on watermelon vine, and (B) on watermelon fruits.
Credit: Mathews Paret UF/IFAS
It is important to note that many of the above-mentioned viruses rarely occur alone and that multiple infections are common.
As of this writing, no cucurbit varieties resistant to these viruses are available. Management of the whitefly vector and weed hosts of the virus is the only option for reducing losses at this time. Whitefly resistance to insecticides is a serious problem with a number of products no longer effective for killing adult whiteflies. Whitefly nymphs are easier to control than adults; this is because adults feed on more than 600 different host plants that include weeds and other cultivated crops. Cultural controls, as outlined below, are essential for a strong resistance management program.
Most of the following management recommendations are taken directly from "Management of Whiteflies, Whitefly-Vectored Plant Virus, and Insecticide Resistance for Tomato Production in Southern Florida," specifically the recommendations for managing Tomato yellow leaf curl virus (TYLCV) and for delaying the development of resistance to insecticides in the whitefly vector (Schuster et al. 2007). Those sections not relevant to cucurbits have been deleted, and some information has been added. The recommendations for tomatoes have evolved over several years and continue to be revised. We envision a similar process for the recommendations for cucurbits.
Crop Hygiene
Field hygiene should be a high priority and should be included as an integral part of the overall strategy for managing whitefly populations, reducing incidence of whitefly-transmitted viruses, and reducing the incidence of insecticide resistance. These practices will help delay the initial whitefly infestation and delay the introduction of viruses into the crop.
Establish a minimum 2-month crop-free period during the summer, preferably from mid-June through mid-August or longer.
Delay planting new fall crops as long as possible and remove spring crops as early as possible to increase the summer crop-free period and avoid carryover of disease and pests to the fall crop.
Try to eliminate, as much as is practical, any cucurbit weeds (balsam apple, creeping cucumber, citron) that could serve as a source of viruses and whiteflies for the crop.
Separate fall and winter cucurbit crops in time and space.
Do not plant new crops near or adjacent to old, infested crops as whitefly adults tend to migrate from old declining crops toward new crops with emerging leaves.
If the cucurbit plants are to be a double crop, especially following tomatoes, the previous or primary crop should be thoroughly destroyed to reduce the initial whitefly population.
Destroy the crop and existing whiteflies quickly and thoroughly.
Promptly and efficiently destroy all vegetable crops within 5 days of final harvest to maximally decrease whitefly numbers and sources of plant viruses.
Use a contact desiccant ("burn down") herbicide in conjunction with a heavy application of oil (not less than 3% emulsion) and a non-ionic adjuvant to destroy crop plants and to quickly kill whiteflies.
Time burn down sprays to avoid windy periods, especially when prevailing winds are blowing whiteflies toward adjacent plantings.
Destroy crops block by block as harvest is completed rather than waiting and destroying the entire field at one time.
Other Cultural Control Practices
Use proper pre-planting cultural practices to avoid introducing whiteflies or viruses into crops.
Grow and/or plant whitefly and virus-free transplants.
When possible, plant cucurbits at least 1.5 mile upwind from key sweet potato whitefly hosts such as tomatoes, soybean, and cotton.
Do not grow, work with, or manipulate vegetable transplants and ornamental plants (i.e., hibiscus, poinsettia) at the same location and at the same time, especially if bringing in plant materials from other areas of the United States or outside the United States.
If this is not possible, isolate vegetable transplants and ornamental plants from each other.
Avoid yellow clothing or implements, which will attract whitefly adults.
Cover all vents and other openings with whitefly-resistant screening (0.25 x 0.8 mm openings or less for passive ventilation, less for forced-air ventilation). Use double doors with positive pressure. Cover roofs with ultraviolet light-absorbing films (UV-absorbing films), or UV-reflecting covers when feasible.
Covering the ground outside the greenhouse or planthouse, particularly adjacent to greenhouse cooling pads, with UV-reflective mulch can reduce entry of whiteflies, aphids, and thrips. See https://edis.ifas.ufl.edu/publication/in730 for this and other details of exclusion practices.
If possible, do not purchase transplants produced in Texas, the southwestern United States, or Mexico, where whitefly-transmitted viruses are common.
Use UV-reflective (aluminum) mulch on plantings that are historically most susceptible to whitefly infestation. Reflective mulches with a black strip (not metalized) down the center to allow the soil to warm will be less likely to delay growth for crops grown in cooler months. The effects of the mulch will be lost when plants cover the mulch.
Incorporate plants such as sweet alyssum (tall white) and African marigold (crackerjack) that are known to be attractive to beneficial insects and pollinators but are not susceptible to whiteflies.
Use proper post-planting practices.
Removal of the earliest plants with symptoms may be of some value, especially for reducing the spread of viruses when whitefly populations are low.
Manage weeds within crops to minimize interference with spraying and to eliminate weeds that can serve as whitefly and virus hosts.
Insecticidal Control Practices
Use a proper whitefly insecticide program that rotates modes of action to delay the development of resistance to insecticides, particularly neonicotinoid insecticides. Read and follow the label; many insecticides have limits on the number of applications that can be made in a growing season.
Insecticides that disrupt whitefly feeding should be privileged as they significantly reduce the transmission of virus. These include diamides, neonicotinoids and pyrifluquinazon (Castle et al. 2017).
On transplants in the production facility, apply a neonicotinoid one time not more than 7 days before planting.
Use soil applications of neonicotinoids at planting for longer-season crops, such as watermelon, so that there is less chance of affecting bees pollinating the crop. During the rest of the season, avoid spraying neonicotinoids; those can be replaced by diamides or flupyradifurone.
For best control, use a neonicotinoid as a soil drench at transplanting/seeding, preferably in the transplant/establishment water. Soil applications of neonicotinoids through the drip irrigation system are generally less efficient and wasteful of product and thus not recommended.
To preserve the neonicotinoid-free period, do not use split applications of soil drenches of neonicotinoid insecticides (i.e., do not apply at transplanting and then again later).
If foliar applications of a neonicotinoid insecticide (dinotefuran, acetamiprid, and thiamethoxam are labeled for this use on cucurbits) are used instead of soil drenches at transplanting, foliar applications should be restricted to the period before flowering because of toxicity to bees. Do not exceed the maximum active ingredient per season according to the label.
Follow scouting recommendations when using a foliar neonicotinoid insecticide program. Switch to non-neonicotinoid insecticide classes after flowering begins, and do not use any neonicotinoid class insecticides for the remaining cropping period. The use of insect growth regulators (buprofezin, novaluron, pyriproxyfen) to control nymphs is one effective option. However, keep in mind that growth regulators do not control adults, but nymphs only.
When possible, use selective rather than broad-spectrum control products, to conserve bees and natural enemies that enhance biological control. Bees are essential for production of cucurbit crops, and insecticides that are toxic to them should be avoided during flowering or applied in the evening when bees are not foraging.
Use kaolin clay as repellent against whitefly adults in addition to or in alternation with conventional insecticides. Kaolin needs to be applied on the bottom of the leaves for best efficacy, which may not be feasible for crops such as watermelon that grow as a rambling vine. To increase kaolin efficacy, it can be tank-mixed with limonene (5ml/L) and an emulsifier (Johnston et al. 2022).
Pymetrozine (Fulfill) has been shown to reduce virus transmission of SqVYV in greenhouse studies if both the infected (source of virus) and uninfected plants are treated, so it will be of most value in preventing spread once this virus is present in the crop.
Do not apply insecticides on weeds on field perimeters. If you are worried that weeds can host whiteflies and their viruses, remove the weeds instead. Applications of insecticides on weeds can kill insect natural enemies and pollinators (weeds have different flowering times than the main crop).
Apply an effective insecticide to kill whitefly adults prior to any cultural manipulations but apply when bees are not foraging.
Do Unto Your Neighbor As You Would Have Them Do Unto You
Look out for your neighbor's welfare.
This may be a strange or unwelcome concept in the highly competitive vegetable industry, but it is in your best interest to remember—and work to protect—neighboring growers and neighboring crops. Growers must remember that should the whiteflies develop full-blown resistance to insecticides, especially the neonicotinoids, it is not just the other guy who will be hurt—everybody will feel the pain!
Know what is going on in the neighbor's fields.
Growers should try to keep abreast of operations in upwind fields, especially harvesting and crop destruction, which both disturb the foliage and cause whitefly adults to fly. Many crops are hosts for whiteflies, and some of them can be infected by the same viruses that infect cucurbits (e.g., beans and CuLCrV).
Adkins, S., S. E. Webb, D. Achor, P. D. Roberts, and C. A. Baker. 2007. "Identification and Characterization of a Novel Whitefly-Transmitted Member of the Family Potyviridae Isolated from Cucurbits in Florida." Phytopathology 97:145–154. https://doi.org/10.1094/PHYTO-97-2-0145
Adkins, S., S. E. Webb, C. A. Baker, and C. S. Kousik. 2008. "Squash Vein Yellowing Virus Detection Using Nested Polymerase Chain Reaction Demonstrates That the Cucurbit Weed Momordica charantia Is a Reservoir Host." Plant Disease 92:1119–1123. https://doi.org/10.1094/PDIS-92-7-1119
Adkins, S., C. G. Webster, C. A. Baker, R. Weaver, E. N. Rosskopf, and W. W. Turechek. 2009. "Detection of Three Whitefly-Transmitted Viruses Infecting the Cucurbit Weed Cucumis melo var. dudaim in Florida." Plant Health Progress 10:1. https://doi.org/10.1094/PHP-2009-1118-01-BR
Akad, F., S. Webb, T. W. Nyoike, O. E. Liburd, W. Turechek, S. Adkins, and J. E. Polston. 2008. "Detection of Cucurbit leaf crumple virus in Florida Cucurbits." Plant Disease 92:648–648. https://doi.org/10.1094/PDIS-92-4-0648C
Brown, J. K., J. C. Guerrero, M. Matheron, M. Olson, and A. M. Idris. 2007. "Widespread Outbreak of Cucurbit yellow stunting disorder virus in Melon, Squash, and Watermelon Crops in the Sonoran Desert of Arizona and Sonora, Mexico." Plant Disease 91:773. https://doi.org/10.1094/PDIS-91-6-0773A
Castle, S., J. Palumbo, P. Merten, C. Cowden, and N. Prabhaker. 2017. "Effects of Foliar and Systemic Insecticides on Whitefly Transmission and Incidence of Cucurbit yellow stunting disorder virus." Pest management science 73 (7): 1462–1472. https://doi.org/10.1002/ps.4478
Hendricks, K., R. N. Hernandez, P. D. Roberts, T. Isakeit, and O. J. Alabi. 2021. "First Report of Watermelon crinkle leaf-associated virus 1 (WCLaV-1) and WCLaV-2 in Watermelon (Citrullus lanatus L.) Plants Co-Infected with Cucurbit chlorotic yellows virus in Florida." Plant Disease 106 (1): 339. https://doi.org/10.1094/PDIS-06-21-1141-PDN
Hochmuth, Robert C., and Richard K. Sprenkel. 2008. "Exclusion Methods for Managing Greenhouse Vegetable Pests: ENY-846/IN730". EDIS 2008 (3). https://doi.org/10.32473/edis-in730-2008
IRAC (Insecticide Resistance Action Committee) Website http://www.irac-online.org
Jailani, A. A. K., F. Iriarte, R. Hochmuth, S. M. Willis, M. W. Warren, K. K. Dey, M. Velez-Climent, J. McVay, S. Bag, and M. L. Paret. 2021. "First Report of Cucurbit chlorotic yellows virus Affecting Watermelon in USA." Plant Disease 106 (2): 774. https://doi.org/10.1094/PDIS-03-21-0639-PDN
Johnston, N., T. Paris, M. Paret, J. Freeman, and X. Martini. 2022. "Repelling Whitefly (Bemisia tabaci) Using Limonene-Scented Kaolin: A Novel Pest Management Strategy." Crop Protection 155:105905. https://doi.org/10.1016/j.cropro.2022.105905
Kuo, Y. -W., M. R. Rojas, R. L. Gilbertson, and W. M. Wintermantel. 2007. "First Report of Cucurbit yellow stunting disorder virus in California and Arizona, in Association with Cucurbit leaf crumple virus and Squash leaf curl virus." Plant Disease 91 (3): 330. https://doi.org/10.1094/PDIS-91-3-0330B
Polston, J. E., L. L. Hladky, F. Akad, and W. M. Wintermantel. 2008. "First Report of Cucurbit yellow stunting disorder virus in Cucurbits in Florida." Plant Disease 92 (8): 1251. https://doi.org/10.1094/PDIS-92-8-1251B
Schuster, David J., Philip A. Stansly, Jane E. Polston, Phyllis R. Gilreath, and Eugene McAvoy. 2007. "Management of Whiteflies, Whitefly-Vectored Plant Virus, and Insecticide Resistance for Vegetable Production in Southern Florida: ENY-735/IN695." EDIS 2007 (17). https://doi.org/10.32473/edis-in695-2007
Wisler, G. C. and J. E. Duffus. 2001. "Transmission Properties of Whitefly-Borne Criniviruses and Their Impact on Virus Epidemiology." In: Virus-Insect-Plant Interactions, edited by K. F. Harris, O. P. Smith and J. E. Duffus. Academic Press, San Diego, CA, USA, pp 293–308. https://doi.org/10.1016/B978-012327681-0/50018-2
Webb, Susan E., F. Akad, Teresia W. Nyoike, Oscar E. Liburd, and Jane E. Polston. 2007. "Whitefly-Transmitted Cucurbit Leaf Crumple Virus in Florida: ENY-477/IN716." EDIS 2007 (11). https://doi.org/10.32473/edis-in716-2007
Publication #ENY-478
Whidden, Alicia J
Webb, Susan E
Stansly, Phillip A
Schuster, David J
Roberts, Pamela D.
Polston, Jane Elizabeth
Paret, Mathews L.
Nyoike, Teresia W
McAvoy, Eugene
Martini, Xavier
Liburd, Oscar E.
Baker, Carlye A
Adkins, Scott
Entomology and Nematology
Cucurbit Diseases
Cucurbit Pest Insects
Insecticide Resistance Management
DOI: doi.org/10.32473/edis-in871-2022
Program Area: ==Agriculture/Horticulture==
This document is ENY-478, one of a series of the Entomology and Nematology Department, UF/IFAS Extension. Original publication date July 2011. Revised June 2015, January 2019, and July 2022. Susan E. Webb, David J. Schuster, Philip A. Stansly, Scott Adkins, Carlye A. Baker, Eugene McAvoy, and Alicia Whidden contributed to earlier versions of this publication. Visit the EDIS website at https://edis.ifas.ufl.edu for the currently supported version.
Xavier Martini, assistant professor, Entomology and Nematology Department, UF/IFAS North Florida Research and Education Center; Mathews Paret, associate professor, Plant Pathology Department, Jane E. Polston, emeritus professor, Plant Pathology Department, UF/IFAS NFREC; Scott Adkins, virologist, USDA-ARS; Pamela Roberts, professor, Plant Pathology Department, UF/IFAS SWFREC; Teresia Nyoike, project leader, BASF Corporation, Research Triangle Park, North Carolina; and Oscar E. Liburd, professor, Entomology and Nematology Department; UF/IFAS Extension, Gainesville, Florida 32611.
Jane Polston
Xavier Martini


text:
\section{Introduction}
\label{sectintro}
\subsection{Vassiliev invariants}
In the year 1990, V.~A.~Vassiliev introduced \cite{Va} a new type of knot invariants that
include the information of most of the invariants that followed the
celebrated discovery of the Jones polynomial (\cite{Jo}, \cite{H}, \cite{Ka}).

An immersion of the circle $S^1$ in the three-sphere $S^3$ having exactly
$m$ double points and no other singularities is called $m$-singular.
Let ${\cal K}_m$ denote the set of ambient isotopy classes of $m$-singular
immersions. The elements of $K_0$ are classes of embeddings, i.e. knots in
the classical sense.

Any knot invariant $v$ with values in an abelian group,
can be extended to singular knots inductively,
by use of the desingularisation rule:
$$v(K) := v(K_+) - v(K_+)$$
where $K \in {\cal K}_m$, $K_+,K_- \in {\cal K}_{m-1}$ differ only locally like this:
\cepsfig{fig_des.ps}

\begin{definition}
\rm
A knot invariant with values in an abelian group is called a
{\it Vassiliev invariant} of degree $m$ iff it vanishes on ${\cal K}_{m+1}$ and the
unknot but not on ${\cal K}_m$.
A Vassiliev invariant $v$ is called {\it primitive} iff it is a monoid homomorphism,
i.e.~$v(K_1\#K_2) = v(K_1)+v(K_2)$, where $\#$ denotes the connected sum operation
for knots.
\end{definition}

\noindent
For a commutative ring $k$ let ${\cal V}_m^k$ (${\cal PV}_m^k$) denote the $k$-module generated
by all $k$-valued (primitive) Vassiliev invariants of degree $\leq m$.

The product of two invariants $v_1,v_2$ is given pointwise by
$v_1\!\!\cdot\! v_2(K) := v_1(K)v_2(K)$ for all $K \in {\cal K}_0$.
It is not hard to show that if the degrees of $v_1, v_2$ are $m_1, m_2$, then
the degree of $v_1\!\!\cdot\! v_2$ is $m_1 + m_2$.
This establishes a graded algebra structure on $\oplus \,{\cal V}_m / {\cal V}_{m-1}$.
It is even a graded Hopf algebra (the coproduct corresponds to the connected sum operation),
which explains why we restrict ourselves to primitive Vassiliev invariants.
\begin{remark}
\label{remprim}
Every Vassiliev invariant can be expressed (uniquely up to invariants
of lower degree) as a polynomial in primitive Vassiliev invariants.
\end{remark}

\noindent
Vassiliev invariants have been defined topologically but they are
closely related to purely combinatorial objects, which we shall describe now.

\subsection{Modules of diagrams}
\begin{definition}
\rm
\begin{enumerate}
\item
A {\it free diagram} (or Chinese character) of degree $(m,u)$ is a finite abstract graph with
$2m-u$ trivalent and $u$ univalent vertices.
The trivalent vertices are rigid, i.e.~a cyclic ordering
of the three arriving edges is chosen at every trivalent vertex.
\item
A diagram together with a linear ordering of its univalent vertices
is called {\it fixed diagram}.
\item
A diagram of degree $(m,u)$ with $u > 0$ together with a cyclic ordering of its univalent vertices
is called {\it circle diagram} of degree $m$.
\end{enumerate}
\end{definition}

\noindent
In all our pictures the edges are ordered counterclockwise at each vertex.
In circle diagrams we depict the cyclic ordering of the univalent vertices
by gluing them on an oriented circle.
We will need four types of local relations (only the changed parts
of the diagrams are shown):
\begin{itemize}
\item
AS (antisymmetry of vertices): \lepsfig{fig_as.ps}
\item
IHX relation: \lepsfig{fig_ihx.ps}
\item
STU relation: \lepsfig{fig_stu.ps}
\item
FI (framing independence): \lepsfig{fig_fi.ps}
\end{itemize}

\hspace{\fill}\\
AS and IHX are homogenous with respect to $m$ and $u$, the STU-relation only
with respect to $m$. The STU and FI relations are defined only for circle diagrams.

\begin{definition}
\rm
We have the following ${\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z}$-modules\footnote{ We will denote the free $k$-module with basis $S$ by $k\langle S \rangle$.}:
\begin{eqnarray*}
{\cal A}_m &:=& \Zmodule {\mbox{ circle diagrams of degree } m\; } \;/\;
\Zmodule {\mbox { STU relations }} \\
{\cal A}_m^{\rm r} &:=& {\cal A}_m / \Zmodule {\mbox { FI relations }} \\
\P_m &:=& \mbox {submodule of } {\cal A}_m \mbox{ generated by connected circle diagrams } \\
\P &:=& \bigoplus_{m=2}^\infty\limits \P_m \\
{\cal B}_{m,u} &:=& \Zmodule {\mbox{ connected free diagrams of degree } (m,u)\; } \;/\;
\Zmodule {\mbox { AS,IHX relations }} \\
F(u) &:=& \Zmodule {\mbox{ conn.~fixed diagrams with } u \mbox{ univalent vertices }} \;/\;
\Zmodule {\mbox { AS,IHX relations }} \\
\end{eqnarray*}
\end{definition}

\noindent
The most important (and highly non-trivial) facts about Vassiliev
invariants may be summarized in the following manner.
\begin{theorem}(Bar-Natan, Birman-Lin, Drinfeld, Kontsevich, Vassiliev)
\begin{eqnarray*}
&{\cal V}_m^{\mathbf{Q}} \;/\; {\cal V}_{m-1}^{\mathbf{Q}} {\;\;\cong\;\;} {\cal A}_m^{\rm r} \otimes {\mathbf{Q}} & \mbox { for }m \geq 1 \\
&{\cal PV}_m^{\mathbf{Q}} \;/\; {\cal PV}_{m-1}^{\mathbf{Q}} {\;\;\cong\;\;} \P_m \otimes {\mathbf{Q}} {\;\;\cong\;\;} \bigoplus_{u=1}^m\limits\; {\cal B}_{m,u}\otimes{\mathbf{Q}} &\mbox{ for } m \geq 2 \\
\end{eqnarray*}
\end{theorem}
\begin{remark}
\rm
To be more specific, there is a natural way to define a map
${\cal V}_m^{\mathbf{Q}} / {\cal V}_{m-1}^{\mathbf{Q}} \rightarrow {\rm Hom}_{\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z}^{} ({\cal A}_m^{\rm r},{\mathbf{Q}})$,
which turns out to be the desired isomorphism. ${\cal A}_m, \P_m, {\cal B}_{m,u}$
are finite dimensional, so we use them instead of their duals.
\end{remark}

\noindent
It is very annoying that our knowledge of $\P$
is so limited.
Dror Bar-Natan has computed ${\rm rk\ } \P_m$ for $m \leq 9$.
Upper and lower bounds for all degrees have been found (\cite{CD1}, \cite{CD2}, \cite{St}) but they
are unacceptably bad. And we know practically nothing about torsion in $\P$.

The main goal of this paper is to describe two algorithms that give
upper bounds for the rank of $\P_m$.
But first, we present a very good lower bound that is due
to Dror Bar-Natan and an algebra $\Lambda$, introduced by Pierre Vogel,
that acts on $\P$.

\subsection{Marked surfaces}
\label{sectmark}
If every edge of a free diagram is labeled with exactly one of the symbols "=" or "x",
it is called a {\it marked diagram}. A {\it marked surface} is a closed
compact surface with some points marked on its boundary.
At each marked point, an orientation of the boundary component is specified.
A marked surface $F$ is {\it normalized}, if either $F$ is orientable and
all markings induce the same orientation on $F$, or $F$ is non-orientable and
the orientations of the markings coincide on each component of $\partial F$.
We call a diagram {\it embedded} if it is drawn on the 2-sphere $S^2$ and
the cyclic order given at each vertex is compatible with the orientation of $S^2$.

We will thicken the five building blocks of embedded marked diagrams (univalent vertices,
trivalent vertices, edges with "=", edges with "x", crossings of edges) like this:
\cepsfig{fig_ms.ps}

\noindent
This assigns to every embedded marked diagram $D$ a marked surface $\hat D$.
If $D$ has $u$ univalent vertices then $\hat D$ has $u$ markings on its boundary.
If $D^\prime$ is the diagram that is obtained after all the markings of
a marked diagram $D$ are forgotten, we say that $D$ is a {\it marking of} $D^\prime$.
Let $x(D)$ denote the number of "x"-marked edges of $D$.

\noindent
Now we can define the "thickening map" $\Phi_m$ from $\bigoplus_{u=1}^{m}{\cal B}_{m,u} \rightarrow \Zmodule{\mbox{marked surfaces}}$:
\begin{definition}
\rm
If an element $b$ of ${\cal B}_{m,u}$ is represented by a embedded diagram $D_b$ then
we define $\Phi_m(b) := \sum_{{\mbox{\tiny all markings}} \atop D {\rm\; of\; } D_b}\limits (-1)^{x(D)}\hat D$.
\end{definition}

\begin{remark}
\rm
It is easy to show that $\Phi_m$ is well-defined, e.g. it does not depend on the
choice of the embedding and it respects the relations AS and IHX. This implies
that ${\rm rk\ } ({\rm im\ }\Phi_m)$ is a lower bound for ${\rm rk\ } \P_m$.
\end{remark}

\noindent
Let $p$ denote the projection from $\Zmodule{\mbox{marked surfaces}}$
onto $\Zmodule{\mbox{normalized marked surfaces}}$ and let $\tilde\Phi_m := p\circ\Phi_m$.
Of course we then have ${\rm rk\ } ({\rm im\ }\tilde\Phi_m) \leq {\rm rk\ } \P_m$ as well.
It can be shown that $\tilde\Phi_m$ contains essentially the same
information as $\Phi_m$.

\begin{remark}
\label{remlie}
\rm
On one side, to every marked surface $s$, there is naturally associated a linear
form on $\P_m$, namely the coefficient of $s$ in $\Phi_m$.
On the other side, to a finite dimensional Lie algebra, equipped with
a symmetric, Ad-invariant, non-degenerate, bilinear form and a finite
dimensional representation, there is associated a linear form on $\P_m$, too.

\noindent
Bar-Natan has shown in \cite{BN1} that all the linear forms obtained via marked surfaces
are coming from Lie algebras in the families so and gl and all of their
representations. He has also shown that the corresponding Vassiliev invariants
contain the same information as the HOMFLY and the ($2$-variable) Kauffman
polynomials and all of their cablings.
\end{remark}

\subsection{The algebra $\Lambda$}

Vogel defined an interesting submodule $\Lambda$ of the module $F(3)$ of fixed diagrams with
three univalent vertices.
The symmetric group $S_u$ acts on $F(u)$ by permutation of the univalent
vertices. There are maps $\phi_i$ ($1 \leq i \leq u$) from $F(u)$ to $F(u+1)$,
given by gluing a trivalent vertex to the $i$-th univalent vertex; the two
new univalent vertices get the numbers $i$ and $i+1$ and the numbers $> i$
are increased by one.

\begin{definition}
\rm
\label{deflambda}
$
\forall u\in F(3): \; u\in\Lambda \;\;:\Leftrightarrow\;\; \sigma(u) = \epsilon(\sigma)u
\mbox{ for all }\sigma\in S_3\;\mbox{ and }\phi_1(u) = \phi_2(u)$,
where $\epsilon$ is the signature homomorphism.
\end{definition}

\noindent
For a diagram $d$ of $\P$ and a diagram $u$ of $\Lambda$ one makes
a simple construction: Delete a trivalent vertex
of $d$ (it always has at least one) and the three univalent vertices
of $u$. The three remaining open edges of $d$ are glued to those
of $u$. The first condition in definition \ref{deflambda} together with the
AS relation cause that all $6$ ways of doing this give the result (in $\P$).
The second condition has the effect that it does not matter, at which trivalent
vertex $u$ is inserted (here it is essential that $d$ is connected).
It is easy to show that the insertion is compatible with the IHX relation,
so $\Lambda$ operates on $\P$.
$\Lambda$ is even a graded algebra because it acts on itself, and
$\P$ is a $\Lambda$-module.

\noindent
It has been shown in \cite{Vo} that $\Lambda\otimes{\mathbf{Q}}$ is commutative and that
it is contained in $\P\otimes{\mathbf{Q}}$:

\begin{proposition}
\label{vogelprop}
$\Lambda\otimes{\mathbf{Q}} {\;\;\cong\;\;} \bigoplus_{m=2}^\infty{\cal B}_{m,2}\otimes{\mathbf{Q}}$
\end{proposition}

\noindent
Furthermore the following elements $t,x_3,x_4,x_5,\ldots$ of $\Lambda$
are constructed:
\cepsfig{fig_vog.ps}

\noindent
Vogel showed that, in degree $\leq 8$,
$\Lambda\otimes{\mathbf{Q}}$ is generated by $t,x_3,x_5,\ldots$ and isomorphic
to ${\mathbf{Q}}[t,x_3,x_5,\ldots]$. He conjectured that this is true in all degrees,
and gave a polynomial in degree $10$ for which he could not tell,
whether it is trivial or not.

\section{Results}
\label{sectres}

We have implemented both algorithms that will be given in section \ref{sectalg} and made
a program that effectively computes the thickening map $\tilde\Phi_m$ described in section
\ref{sectmark}.
The output of these three programs for degree $m$ will be denoted $O_A(m),
O_B(m),O_C(m)$,
respectively. By corollary \ref{corineq} we have that $O_A(m),O_B(m) \geq {\rm rk\ } \P_m \geq O_C(m)$.
First we confirmed for $3 \leq m \leq 9$ that $O_A(m) = O_B(m) = O_C(m) =$ values given in \cite{BN2}.
Then we found $O_A (10) = O_B(10) = O_C(10) = 27$, $O_A(11) = O_B(11) = O_C(11) = 39$.
We were astonished that our algorithms are good enough to give
the exact values. The complete surprise came with $m = 12$;
we had conjectured $O_C(12) \neq {\rm rk\ } \P_{12}$, because of Vogel's conjecture.
After some hundred hours of CPU-time we got the result $O_B(12) = O_C(12) = 55$
(the computation of $O_A(12)$ is too exhaustive and could not be performed).

\subsection{Vassiliev invariants}
\begin{theorem}[computational result]
The sequence of the dimensions of the spaces of rational-valued primitive Vassiliev invariants of degree $m$
modulo those of degree $m-1$ starts
$$0,\;1,\;1,\;2,\;3,\;5,\;8,\;12,\;18,\;27,\;39,\;55,\;\ldots$$
\end{theorem}

\begin{remark}
{\rm
For some time only the first seven numbers of this sequence were known and there
was some excitement, because it appeared to be the famous Fibonacci
sequence. It is a somehow mysterious coincidence that it
is again a Fibonacci number in degree twelve.
}
\end{remark}

\noindent
In view of remark \ref{remlie} we have the following consequence of $O_C(m) = {\rm rk\ } \P_m$
for $m \leq 12$.

\begin{corollary}
All rational Vassiliev invariants of degree $\leq 12$ are coming from representations
of the classical Lie algebras so and gl.
\end{corollary}

\begin{remark}
{\rm
Vogel has shown that a similar statement is false for sufficiently large
degrees. Let $m_c$ denote the minimal degree for which not
all Vassiliev invariants are coming from semi-simple Lie algebras.
Results of Jens Lieberum (\cite{Li}) and our calculations together imply that
$13 \leq m_c \leq 17$.}
\end{remark}

\begin{corollary}
Vassiliev invariants up to degree twelve can not distinguish knots from
their inverses.
\end{corollary}
{\bf Proof\quad}
It has been shown that Vassiliev invariants coming from semi-simple
Lie algebras are orientation insensitive. Another way is to
verify that ${\rm rk\ } {\cal B}_{m,u} = 0$ for $m \leq 12$ and $u$ odd (see table
in section \ref{secttab}).
\qed

\hspace{\fill}\\
A statement about ${\cal A}$ similar to remark \ref{remprim} makes it is
easy to compute the ranks of $A_m^{\rm r}$ and $A_m$,
corresponding to the numbers of Vassiliev invariants of knots
and Vassiliev invariants of framed knots, if
${\rm rk\ } \P_m$ is known.

\hspace{\fill}\\
\centerline{\begin{tabular}{||c|c|c|c|c|c|c|c|c|c|c|c|c|c||}
\hline
$m$ &
	$0$ &   $1$ &   $2$ &   $3$ &   $4$ &   $5$ &   $6$ &   $7$ &
	$8$ &   $9$ &   $10$ &  $11$ &   $12$  \\
\hline
${\rm rk\ }\P_m$ &
	$0$ &   $1$ &   $1$ &   $1$ &   $2$ &   $3$ &   $5$ &   $8$ &
	$12$ &  $18$ &  $27$ &  $39$ &   $55$ \\
${\rm rk\ }{\cal A}_m$ &
	$1$ &   $1$ &   $2$ &   $3$ &   $6$ &   $10$ &   $19$ &   $33$ &
	$60$ &  $104$ & $184$ &  $316$ &   $548$ \\
${\rm rk\ }{\cal A}_m^{\rm r}$ &
	$1$ &   $0$ &   $1$ &   $1$ &   $3$ &   $4$ &   $9$ &   $14$ &
	$27$ &  $44$ &  $80$ &  $132$ &   $232$   \\
\hline
\end{tabular}}

\subsection{Results about $\P$ and $\Lambda$}
\label{secttab}

Our programs work over the field ${\mathbf{F}}_2$, so due to corollary \ref{cortorsion}
we have a little statement about torsion in $\P$.

\begin{corollary}
There is no $2$-torsion in $\P$ in degree $\leq 12$.
\end{corollary}

\noindent
By counting the dimensions of the image of the thickening map $\Phi$ for
each ${\cal B}_{m,u}$ separately, we get the following table for ${\rm rk\ } {\cal B}_{m,u}$.

\hspace{\fill}\\
\centerline{\begin{tabular}{||c|c|c|c|c|c|c|c|c||}
\hline
${\rm rk\ } {\cal B}_{m,u}$&       $u=2$ & $u=4$ & $u=6$ & $u=8$ & $u=10$ & $u=12$ & total \\
\hline
$m=1$ & $1$ &   &       &       &       &       &       $1$ \\
\hline
$m=2$ & $1$ &   &       &       &       &       &       $1$ \\
\hline
$m=3$ & $1$ &   &       &       &       &       &       $1$ \\
\hline
$m=4$ & $1$ &   $1$ &   &       &       &       &       $2$ \\
\hline
$m=5$ & $2$ &   $1$ &   &       &       &       &       $3$ \\
\hline
$m=6$ & $2$ &   $2$ &   $1$ &   &       &       &       $5$ \\
\hline
$m=7$ & $3$ &   $3$ &   $2$ &   &       &       &       $8$ \\
\hline
$m=8$ & $4$ &   $4$ &   $3$ &   $1$ &   &       &       $12$ \\
\hline
$m=9$ & $5$ &   $6$ &   $5$ &   $2$ &   &       &       $18$ \\
\hline
$m=10$ & $6$ &   $8$ &   $8$ &   4 &  $1$ &     &       $27$ \\
\hline
$m=11$ & $8$ &   $10$ &  $11$ &  $8$ & $2$ &     &      $39$ \\
\hline
$m=12$ & $9$ &   $13$ &  $15$ &  $12$ & $5$ & $1$ &     $55$ \\
\hline
\end{tabular}}

\begin{corollary}
The algebra morphism from ${\mathbf{Q}}[T,X_3,X_5,\ldots]$ to
$\Lambda\otimes{\mathbf{Q}}$ given by $T \rightarrow t$, $X_i \rightarrow x_i$ is
not an isomorphism. In degree $< 11$ it is surjective and has a one dimensional kernel
(living in degree $10$).
\end{corollary}
{\bf Proof\quad}
By calculating characters, Vogel has already shown that this algebra
morphism is injective in degree $\leq 9$ and its kernel in degree $10$ has at
most dimension one. Because of proposition \ref{vogelprop}, the statements can be verified
by counting the number of monomials in each degree $m$ and comparing it
to ${\rm rk\ } {\cal B}_{m+2,2}$.
In degree $10$, for example, there are ten monomials ($t^{10},t^7x_3^{},t^5x_5^{},t^3x_7^{},
tx_9^{},t^4x_3^2,t^2x_3^{}x_5^{},x_3^{}x_7^{},x_5^2$ and $tx_3^3$), but we
have ${\rm rk\ } {\cal B}_{12,2} = 9$.
\qed

\hspace{\fill}\\
So one half of Vogel's conjecture is false, but the calculations show that up
to degree $10$ the other one holds:

\hspace{\fill}\\
{\bf Conjecture\quad}
$\Lambda\otimes{\mathbf{Q}}$ is generated (as algebra over ${\mathbf{Q}}$) by the elements $t,x_3,x_5,\ldots$

\begin{corollary}
$\P\otimes{\mathbf{Q}}$ is not a free $\Lambda\otimes{\mathbf{Q}}$-module.
\end{corollary}
{\bf Proof\quad}
Let us assume that ${\cal B}_{m,4}\otimes{\mathbf{Q}}$ is a free $\Lambda\otimes{\mathbf{Q}}$-module
with rank $\alpha_m \geq 0$ ($m \geq 4$). Let $\lambda_m$ denote the dimension
of $\Lambda\otimes{\mathbf{Q}}$ in degree $m$. Then we have the following
formula for the rank of ${\cal B}_{m,4}$:
$${\rm rk\ } {\cal B}_{m,4} = \dim_{\mathbf{Q}} {\cal B}_{m,4}\otimes{\mathbf{Q}} = \sum_{i=4}^{m} \lambda_{m-i}\alpha_{i}$$
We have $\lambda_0,\ldots,\lambda_7 = 1,1,1,2,2,3,4,5$,
which together with the values ${\rm rk\ } B_{4,4},\ldots,{\rm rk\ } {\cal B}_{11,4}$
implies $\alpha_4 = \alpha_6 = \alpha_8 = \alpha_{10} = 1$,
$\alpha_5 = \alpha_7 = \alpha_9 = 0$, $\alpha_{11} = -1$.

This contradiction shows that $-$ at least in the $u = 4$ column
of $\P\otimes{\mathbf{Q}}$ $-$ nontrivial relations hold.\qed

\hspace{\fill}\\
We have found another relation which is located in the $u=6$ column in degree $12$.
Unlike here, its existence can not be shown by simply counting dimensions.

\subsection{The structure of $\P\otimes{\mathbf{Q}}$ as far as we know it}

Using the thickening map, we have built a minimal set of diagrams ${\Omega_{^{_{\leq 12}}}^{}}$
that generate $\P\otimes{\mathbf{Q}}$ as $\Lambda\otimes{\mathbf{Q}}$-module in degrees up to $12$.
We were trying to make the elements of ${\Omega_{^{_{\leq 12}}}^{}}$ as simple as possible and
finally, this lead us to a very special type of diagrams:
\begin{definition}
Let $\omega_{i_1i_2\ldots i_k}$ denote the element of ${\cal B}_{i_1+\ldots+i_k+k-1,\;i_1+\ldots+i_k}$
that is represented by a "caterpillar" diagram consisting
of $k$ "body segments" with $i_1,\ldots,i_k$ "legs", respectively.
\end{definition}
Here are some examples of caterpillar diagrams for $\omega_4,\;\omega_{302},\;\omega_{13},\;\omega_{02131}$.
\cepsfig{fig_cp.ps}

\begin{remark}
\label{remcat}
\rm
It is a nice exercise to use the AS and IHX relations to prove that $\omega_{i_1i_2\ldots i_k}$
is well defined (i.e. for inner segments it makes no difference on which
side of the body the legs are drawn).
An easy consequence is $\omega_{i_1\ldots i_k} = \omega_{i_k\ldots i_1}$.
The diagrams $\omega_i$ are also called "wheels with $i$ spokes".
\end{remark}

\noindent
Let ${\Omega_{^{_{\leq 12}}}^{}}$ denote the set consisting of the following $31$ elements.
\newcommand{\w}[1]{\omega_{#1}^{}}
\newcommand{\oneoftwo}[1]{\vbox{\hbox{$#1$}\vspace{3pt}}}
\newcommand{\twooftwo}[2]{\vbox{\vspace{3pt}\hbox{$#1$}\vspace{3pt}\hbox{$#2$}}}
\newcommand{\threeofthree}[3]{\vbox{\vspace{3pt}\hbox{$#1$}\vspace{3pt}\hbox{$#2$}\vspace{3pt}\hbox{$#3$}}}
\newcommand{\oneofthree}[1]{\vbox{\hbox{$#1$}\vspace{7pt}}}
\newcommand{\twoofthree}[2]{\vbox{\hbox{$#1$}\vspace{3pt}\hbox{$#2$}\vspace{3pt}}}
\noindent
\begin{center}
\begin{tabular}{||c|c|c|c|c|c|c||}
\hline
${\Omega_{^{_{\leq 12}}}^{}}_{_{}}$&$ u=2 $&$ u=4 $&$ u=6 $&$ u=8 $&$ u=10 $&$ u=12 $\\
\hline
$m=2 $&$ \w{2} $&$  $&$  $&$  $&$  $&$  $\\
\hline
$ m=3 $&$  $&$  $&$  $&$  $&$  $&$  $\\
\hline
$ m=4 $&$  $&$ \w{4} $&$  $&$  $&$  $&$  $\\
\hline
$ m=5 $&$  $&$  $&$  $&$  $&$  $&$  $\\
\hline
$ m=6 $&$  $&$ \w{202} $&$ \w{6} $&$  $&$  $&$  $\\
\hline
$ m=7 $&$  $&$  $&$ \w{24} $&$  $&$  $&$  $\\
\hline
$ m=8 $&$  $&$ \w{20002} $&$ \w{204} $&$ \w{8} $&$  $&$  $\\
\hline
$ m=9 $&$  $&$  $&$ \w{2004} $&$ \w{26} $&$  $&$  $\\
\hline
$\oneoftwo{\,\,\,m=10} $&$  $&$ \oneoftwo{\w{2000002}} $&$ \twooftwo{\w{20004}}{\w{20202}} $&$
\twooftwo{\w{206}}{\w{404}} $&$ \oneoftwo{\w{1\!0}}  $&$  $\\
\hline
$\oneofthree{\,\,\,m=11} $&$  $&$  $&$ \oneofthree{\w{200004}} $&$ \threeofthree{\w{2006}}{\w{4004}}{\w{2222}} $&$ \oneofthree{\w{28}}  $&$  $\\
\hline
$\oneofthree{\,\,\,m=12} $&$  $&$ \oneofthree{\w{200000002}} $&$ \twoofthree{\w{2000004}}{\w{2000202}} $&$ \threeofthree{\w{20006}}{\w{40004}}{\w{20204}} $&$ \threeofthree{\w{208}}{\w{406}}{\w{262}} $&$ \oneofthree{\w{1\!2}} $\\
\hline
\end{tabular}
\end{center}

\begin{remark}
\rm
At several places in the upper table the choice of a minimal generating set is not
unique. We tried to make ${\Omega_{^{_{\leq 12}}}^{}}$ look as uniform as possible. At first
place we were able to renounce on $\omega$'s with odd indices. After this
only few choices still had to be done. For example,
we preferred $\omega_{26}$ over $\omega_{44}$ (because of the other entries
of the form $\omega_{2,u-2}$) and $\omega_{2222}$ over $\omega_{2204}$ (because
of its symmetry).
\end{remark}

\noindent
Let $\P_\omega$ denote sub-$\Lambda$-module of $\P$ that is generated by caterpillar diagrams.
A glance through the table ${\Omega_{^{_{\leq 12}}}^{}}$ immediately opens the following two questions:

\begin{enumerate}
\item
Is $\P_\omega$ already generated by the caterpillar diagrams with even indices?
\item
Is $\P_\omega = \P$?
\end{enumerate}
The AS relation and remark \ref{remcat} implies that caterpillar diagrams with an odd number of univalent vertices are always trivial.
So if question 2.~could be answered positively, it would imply that all
Vassiliev invariants are orientation insensitive.

It is tempting to make conjectures about how this table continues (especially
for the $u = 4$ column), but let us just summarize what we know for sure.

\begin{remark}
\label{remom}
\rm
If $\Omega = \bigcup \Omega_{m,u}$ is a minimal set of free diagrams that
generate $\P\otimes{\mathbf{Q}}$ as $\Lambda\otimes{\mathbf{Q}}$-module, then
\begin{itemize}
\item
The $u = 2$ column is essentially empty: $\Omega_{m,2}
= \emptyset$ for $m > 2$.
\item
The $u = m = 2i$ diagonal consists only of wheels: $\Omega_{2i,2i} = \{\;\omega_{2i}\;\}$.
\item
On the first subdiagonal ($m-1 = u = 2i$) we have exactly $\lfloor {i\over 3}\rfloor$ elements.
A natural choice is $\Omega_{2i+1,2i} := \{ \;\omega_{ab} \;\vert\; a > 0;\; a$ even$;\; 2a\leq b;\; a+b = 2i\;\}$.
\item
On the second subdiagonal ($m-2 = u = 2i$) there are exactly $\#\Omega_{2i+2,2i} = \lfloor{(i+1)^2\over 12}+{1 \over 2}\rfloor$ elements.
\item
For odd $u$ we know $\Omega_{m,u} = \emptyset$ if $m-u \leq 5$ or $u = 1$ or $m \leq 12$.
\end{itemize}
\end{remark}
{\bf Proof\quad}
The first statement is due to proposition \ref{vogelprop}. The second is obvious and the third
and forth follow from results of Oliver Dasbach in \cite{Da2}.
He showed that $\Omega_{2i+1,2i}\cup\{\;t\omega_{2i}\;\}$ is a basis
for ${\cal B}_{2i+1,2i}$ and $\dim{\cal B}_{u+2,u}=\lfloor { {u^2+12u} \over {48} }\rfloor+1$
for $u$ even.
It can be verified that $\{ \;t^2\omega_{2i}\;\}\cup \; t\,\Omega_{2i+1,2i}$
are independent in ${\cal B}_{2i+2,2i}$, so $\#\Omega_{2i+2,2i} = \lfloor { {4i^2+24i} \over {48} } \rfloor + 1
- \lfloor {i\over 3}\rfloor - 1 = \lfloor{(i+1)^2\over 12}+{1 \over 2}\rfloor$.
Dasbach has also shown (\cite{Da1}) that ${\cal B}_{m,u} = 0$ for $m-u \leq 5$ and $u$ odd.
The same statement for $u=1$ is easy to prove.
\qed

\begin{remark}
\rm
Some time after having found ${\Omega_{^{_{\leq 12}}}^{}}$, we discovered that caterpillar
diagrams have already been used by Chmutov and Duzhin in \cite{CD2}, who call them
"baguette diagrams". The main theorem of \cite{CD2} states that the elements $\omega_{n_1\ldots n_k}$
with all $n_i$ even, $\sum_{i=1}^{j-1} n_i < n_j$ for $j < k$ and
$\sum_{i=1}^{k-1} n_i < {1 \over 2}n_k$ are linearly independent.
This result is quite striking, but it is useless in our context, because
the first interesting\footnote{ The third item of remark \ref{remom} and
the simple relation $\omega_{0,n_2\ldots n_k} = 2t\omega_{n_2\ldots n_k}$ should make clear
that only the diagrams $\omega_{n_1\ldots n_k}$ with $n_1 > 0$ and $k > 2$
are of further interest.}
diagram in this set is $\omega_{2,4,14}$ and lies in
degree $22$.
\end{remark}

\renewcommand{\w}[1]{w_{#1}^{}}
\newcommand{\x}[2]{X_{#1}^{#2}}
\renewcommand{\t}[1]{T_{}^{#1}}
\newcommand{+}{+}
\newcommand{-}{-}
\noindent
Let $W$ denote the image of ${\Omega_{^{_{\leq 12}}}^{}}$ under the map $\omega_* \rightarrow w_*$
and make the following abbreviations:
{
\scriptsize
\begin{eqnarray*}
P &\!:=\!&\! 32\t{10}\! - 152\t{7}\!\x{3}{} + 252\t{5}\!\x{5}{}
         - 101\t{4}\!\x{3}{2} - 36\t{3}\!\x{7}{}  - 9\t{2}\!\x{3}{}\x{5}{}
         + 14\t{}\x{3}{3} + 9\x{3}{}\x{7}{} - 9\x{5}{2} \\
Q & := & \left(32\t{7}\! + 35\t{4}\!\x{3}{} - 9\t{2}\!\x{5}{} - 4\t{}\x{3}{2}\right)\!\otimes\!\w{4}
       + \left(- 76\t{5}\! + 10\t{2}\!\x{3}{} + 3\x{5}{}\right)\!\otimes\!\w{202}
       + \left(12\t{3}\! - 3\x{3}{}\right)\otimes\w{20002} \\
R & := & \left(- 16\t{6}\! + 21\t{3}\!\x{3}{} - 3\t{}\x{5}{} - 2\x{3}{2}\right)\!\otimes\!\w{6}
       + \left(32\t{5}\! - 17\t{2}\!\x{3}{} + 3\x{5}{}\right)\!\otimes\!\w{24}
       + \left(- 36\t{4}\! + 9\t{}\x{3}{} \right)\!\otimes\!\w{204}\\
&&     + \left(12\t{3}\! - 3\x{3}{}\right)\!\otimes\!\w{2004} \\
\end{eqnarray*}
}
\begin{theorem}
\label{theop}
There is a module morphism
$$\mu \;\;:\;\; Q[T,X_3,X_5,\ldots]\otimes\Qmodule{W} \;\;/\;\;
\Qmodule{P\otimes w_2, Q, TQ, R\;} \;\;\;\rightarrow\;\;\; \P\otimes{\mathbf{Q}}.$$
It is an isomorphism in degree $\leq 12$.
\end{theorem}
{\bf Proof\quad}
$\mu$ is given by $\mu(T^aX_3^bX_5^c\ldots\otimes w_*) = t^ax_3^bx_5^c\ldots \omega_*$.
We have seen that Bar-Natan's thickening map $\Phi$ is injective on $\P_m\otimes{\mathbf{Q}}$
for $m \leq 12$.
So we have to calculate $\Phi$ for all diagrams
of the form $t^ax_3^bx_5^c\ldots \omega_*$ ($\omega_* \in {\Omega_{^{_{\leq 12}}}^{}}$) with
degree $\leq 12$ (there are exactly $175$ of them).
The program and the results are available via the internet
(see section \ref{hintsec}).
It is then easy to verify that the span of the resulting vectors
is $171$ dimensional and that
$P\otimes w_2$, $Q$, $TQ$ and $R$ span the kernel of $\Phi\circ\mu$.
So $\mu$ is well defined, injective, and because of
$\sum_{m=2}^{12}\nolimits {\rm rk\ } \P_m = 171$ it is also surjective. \qed

\section{The principle behind the algorithms}
\label{sectprin}

We will now describe a prototype of an algorithm that yields an upper
bound for the rank (ubr) of a finitely generated abelian group $A$.

\begin{definition}
{\rm
A quintuple $(k,S,\varphi,\delta,\rho)$ where $k$ is a field,
$S$ is a finite set, $\varphi$ is a mapping $\varphi: S \rightarrow A$
and $\delta,\rho$ are endomorphisms of $\module kS$ shall be called
{\it ubr-algorithm} for the finitely generated abelian group $A$, iff the following conditions are satisfied:
{
\setlength{\parskip}{-3pt}
\begin{enumerate}
\setlength{\itemsep}{-2pt}
\item
$\varphi(S)$ is a set of generators of $A$,
\item
there exists a integer $j$ such that $\delta^{j+1} = \delta^{j}$,
\item
$\hat\varphi \circ \delta = \hat\varphi$,
\item
$\hat\varphi \circ \rho = 0$.
\end{enumerate}
}
\noindent
Here $\hat\varphi$ denotes the vectorspace homomorphism
$\hat\varphi: \module kS \rightarrow A\otimes_{\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z}^{}\!k$ that is induced by $\varphi$.
}
\end{definition}

\begin{definition}
\label{defubralg}
{\rm
For a given ubr-algorithm let $\Delta := \delta^j$ and ${\cal I} := {\rm im}\, \Delta$.
Then $\bar\rho := \Delta\circ\rho\vert_{{\cal I}}$ is an ${\cal I}$-endomorphism.
The {\it output} of the ubr-algorithm defined as the natural number
${\rm output}(k,S,\varphi,\delta,\rho) := \dim_k \ker \bar\rho$.
}\end{definition}

\begin{lemma}
\label{helplemma}
Conditions 2 and 3 imply
$\Delta(\ker \hat\varphi) = \ker \hat\varphi \cap {\cal I}$.
\end{lemma}
{\noindent\bf Proof:}
"$\supset$": 2.$\;\Rightarrow\; \Delta$ is a projection onto ${\cal I}$.
\\
$"\subset"$: 3.$\;\Rightarrow\; \hat\varphi \circ\Delta = \hat\varphi
\;\Rightarrow\; \Delta(\ker \hat\varphi) \subset \ker \hat\varphi$.
\qed

\begin{proposition}
If $(k,S,\varphi,\delta,\rho)$ is an ubr-algorithm for the finitely generated
abelian group $A$ then
${\rm output}(k,S,\varphi,\delta,\rho) \geq {\rm rk\ } A$.
\end{proposition}

{\noindent\bf Proof:}
The first condition of definition \ref{defubralg} implies that $\hat\varphi$ is an epimorphism.
Due to 4.~we have $\rho({\cal I}) \subset \ker \hat\varphi$.
Together with lemma \ref{helplemma} and $\hat\varphi \circ\Delta = \hat\varphi$
we get:
\begin{eqnarray*}
\dim_k (A\otimes_{\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z}^{}\! k) &
 = & \dim_k \hat\varphi(\module kS) \;\;=\;\; \dim_k \hat\varphi\circ\Delta (\module kS) \;\;=\;\; \dim_k \hat\varphi({\cal I})\\
&= &\dim_k {\cal I} - \dim_k \ker (\hat\varphi\vert_{{\cal I}})
 \;\;=\;\; \dim_k {\cal I} - \dim_k \Delta (\ker \hat\varphi)\\
&\leq& \dim_k {\cal I} - \dim_k \Delta \big(\rho ({\cal I})\big)
 \;\;=\;\ \dim_k\ker\bar\rho
\end{eqnarray*}
Let $T$ denote the maximum torsion subgroup of $A$. $A$ is isomorphic
to $T\times {\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z}^{{\rm rk\ } A}$, and thus $\dim_k (A\otimes_{\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z}^{}\! k) \geq {\rm rk\ } A$.
\qed

\begin{corollary}
\label{cortorsion}
If $(F_p,S,\varphi,\delta,\rho)$ is an ubr-algorithm for $A$ and
${\rm output}(F_p,S,\varphi,\delta,\rho) = {\rm rk\ }\, A$ then $A$ has no
elements of order $p$.
\end{corollary}

\begin{remark}
\rm
To calculate ${\rm output}(k,S,\varphi,\delta,\rho)$ one has to find a basis of
${\cal I}$ and to evaluate the nullity of the corresponding matrix for $\bar\rho$.
So the costs (in time and space) of an ubr-algorithm mainly depend
on the dimension of $\cal I$, whereas the quality of the result (the sharpness
of the upper bound) depends on the choice of $\rho$.
\end{remark}

\section{Two algorithms for $\P_m$}
\label{sectalg}
\subsection{Circle diagrams without loops}
\label{sectcirc}
Let $S_n$ denote the set of permutations of $n$ elements.
We will use the standard linear ordering for the permutations:
$$\pi < \phi \;\; :\Leftrightarrow \;\; \exists \, i \in \{1,\ldots,n\}: \pi(i) < \phi(i)
\mbox { and } \pi(j) = \phi(j) \mbox{ for all } 1 \leq j < i.$$
Denote by $\tau_i$ the elementary transpositions $(i \;\; i\!+\!1)$ and let the product of
permutations be defined by $(\pi\phi)(i) := \phi(\pi(i))$.

\begin{lemma}
\label{smallerlemma}
For any $\pi \in S_n$ the following two statements hold:
If there exists an integer $i$ $(1 \leq i < n)$ with $\pi(i) > \pi(i+1)$  then
$\tau_i\pi < \pi$.
If there exist $i,j$ $(1 \leq i < j \leq n)$ with $\pi(i) = \pi(j)+1$ then $\pi\tau_{\pi(j)} < \pi$.
\end{lemma}
{\bf Proof:} Simply identify the $i$ in the upper definition of $<$ with this $i$ here.
\qed

\hspace{\fill}\\
\noindent
The {\it picture} of a permutation $\pi \in S_n$ is given by two vertical lines with
$n$ distinct points marked on each, together with $n$ lines connecting the $i$-th and the
$\pi(i)$-th point (counted upwards). For example the picture of $(1\;2\;4)(3)$ is
\cepsfig{fig1.ps}

\noindent
\begin{definition}
{\rm
For any $\pi \in S_n$ we will denote by $D^A_\pi$ the element of ${\cal P}_{n+1}$
that is obtained by replacing the box in the following figure
by the picture of $\pi$.
\cepsfig{fig6.ps}

\noindent
The map $\varphi_A : S_n \rightarrow  \P_{n+1}$ is given by
$\pi \rightarrow D^A_\pi$.
}\end{definition}

\noindent
We will now introduce three types of moves, which replace a permutation by
a linear combination of permutations, by showing the parts of their pictures
that are concerned. Omitted parts are indicated by dots and are assumed to
be identical in a row.

\hspace{\fill}\\
\noindent
Type I
\smallskip
\hspace{\fill}\\
\epsfig{fig2.ps}

\hspace{\fill}\\
\noindent
Type II
\smallskip
\hspace{\fill}\\
\epsfig{fig3.ps}

\hspace{\fill}\\
\noindent
Type III
\smallskip
\hspace{\fill}\\
\epsfxsize = \textwidth \epsfig{fig4_1.ps}
\hspace{\fill}\\
\epsfxsize = \textwidth \epsfig{fig4_2.ps}
\hspace{\fill}\\
\epsfxsize = \textwidth \epsfig{fig5_1.ps}
\hspace{\fill}\\
\epsfxsize = \textwidth \epsfig{fig5_2.ps}

\hspace{\fill}\\
The move I should be interpreted in the following way: when $\pi(n) = n$ then replace $\pi$
by $\tau_1\cdots\tau_{n-1}\,\pi\,\tau_{n-1}\cdots\tau_1$; the move II reads:
when there is some $i \in \{1,\ldots,n-1\}$ with $\pi(i) = \pi(i+1)+1$ then
replace $\pi$ by $\tau_i\,\pi - \tau_1\cdots\tau_{i-1}\,\pi\,\tau_{\pi(i)-1}\cdots\tau_1$; etc.

We call a move {\it reducing}, if all the permutations on the right side are
smaller than the permutation $\pi$ on the left, and we then say that $\pi$ allows
a reduction. Permutations that do not allow any reduction are called irreducible.

If a diagram allows more than one reducing move, we have to make a choice.
To make our calculations reproducible, we describe how our implementation works.
Moves of type II are indexed by the left height of the two crossing
lines that are concerned. Moves of type III are indexed by the height
of the left endpoint of the line that is "switched" during the move.
Now all possible moves of a permutation can be ordered by I $<$ II $<$ III
and II$_i$ $<$ II$_j$ and III$_i$ $<$ III$_j$ for $i < j$.

\begin{definition}
{\rm
\label{defdeltaa}
When $\pi \in S_n$ allows reductions of type I, II or III then set $\delta_A (\pi) := $ result of the
smallest reducing move.
When $\pi$ is irreducible set $\delta_A (\pi) := \pi$.
}\end{definition}

\begin{remark}
\rm
Lemma \ref{smallerlemma} implies that moves I and II are always reducing,
and that for moves of type III it suffices to check the first term on the left right.
\end{remark}

\noindent
In the following, we will work with a special element $\Theta_k$ of ${\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z}[S_n]$:
$$\Theta_k := \prod\limits_{i=1}^{k-1}(1-\prod\limits_{j=1}^{k-i}\tau_{k-j}) =
(1-\tau_{k-1}\tau_{k-2}\cdots\tau_1)(1-\tau_{k-1}\cdots\tau_2)\cdots(1-\tau_{k-1})$$
\begin{definition}
{\rm
\label{defrhoa}
For $\pi\in S_n$ set $k := n+1-\pi^{-1}(1)$ and let $\pi^\prime$ be
given by
$$\pi^\prime(i) = \left\{
\begin{array}{lcl}
\pi(n+1-k+i)-1 &\mbox{ for } & 1 \leq i < k \\
n &\mbox{ for } & i = k \\
\pi(n+1-i)-1 &\mbox{ for } & k < i \leq n\\
\end{array}
\right\} \mbox { (thus }\pi^\prime \in S_n\mbox{).}
$$
Now let $\rho_A$ be the $\Zmodule {S_n}$-endomorphism given via
$\rho_A(\pi) := \pi-(-1)^{n-k}\Theta_k \pi^\prime \in {\cal S}_n$.
}
\end{definition}

\noindent
This completes the description of the first algorithm
$A_m := ({k},S_{m-1},\varphi_A,\delta_A,\rho_A)$.

\subsection{Circle diagrams with one loop}
\label{sectone}

To compare permutations of different symmetric groups, we
extend the definition of "$<$" in the following way: $i < j$, $\pi_1 \in S_i$,
$\pi_2 \in S_j \Rightarrow \pi_1 < \pi_2$.

\begin{definition}
{\rm
For $\pi \in S_n$ let $D^B_\pi$ be the element of ${\cal P}_{n}$ that is obtained by replacing
the box in the following figure by the picture of $\pi$.
\cepsfig{fig6l.ps}

\noindent
Let $\varphi_{B,m}$ denote mapping $\bigcup_{n=3}^{m} S_n \rightarrow \P_m$
that is given by $S_n \ni \pi \rightarrow t^{m-n}D^B_\pi$.
}
\end{definition}

\noindent
To define $\delta_B$ we have to modify the first two moves.
The moves of type III given in section \ref{sectcirc}
and the ordering of moves can be adopted unchanged.

\hspace{\fill}\\
Let $\mu_n,\nu_n \in S_n$ be permutations given by $\mu_n(k)=n+1-k$ and
$\nu_n = (1\;\;2\;\ldots\;n)$.
$G := {\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z}/n{\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z} \times {\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z}/2{\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z} \times {\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z}/n{\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z}$ acts on $S_n$ by $(a,b,c)\pi := \nu_n^a\mu_n^b\pi\nu_n^c$
for all $\pi \in S_n$.
For $\pi\in S_n$ let $(\alpha,\beta,\gamma)$ denote an element of $G$ such that
$(\alpha,\beta,\gamma)\pi$ is minimal in the orbit $G\pi$ of $\pi$.
The move I$^\prime$ is to replace $\pi$ by $(-1)^{n\beta}(\alpha,\beta,\gamma)\pi$.

\hspace{\fill}\\
The move II$^\prime$ is given graphically:
\cepsfig{fig3_.ps}

\noindent
If the move II$^\prime$ is applied to an element of $S_n$, the second term
in the result lies in $S_{n-1}$. We will use this move only for $n \geq 4$, so
we do not have to deal with the symmetric groups $S_1$ or $S_2$.

\begin{definition}
{\rm
\label{defdeltab}
When $\pi \in S_n$ allows reductions of type I$^\prime$, II$^\prime$ or III then set $\delta_B (\pi) := $ result of the
smallest reducing move.
When $\pi$ is irreducible let $\delta_B (\pi) := \pi$.
}\end{definition}

\noindent
For any $\pi \in S_{n-1}$ and $0 \leq k \leq n-3$ we have the following
element of ${\mathbf{Z}}} %% {{\sf Z\hspace*{-0.9ex}Z}[S_n]:$
$$\Upsilon_{n,k}^{} (\pi) := \chi_{1,n-k-2}^{} \; \Theta_{n-k-2}^{} \; \chi_{n-k-2,k+1}^{} \; (1 - \tau_n^{})\; \pi^\#$$
Here $\Theta_{k}^{}$ is the same as in section \ref{sectcirc} and
$\chi_{r,s}^{},\pi^\# \in S_n$ are given by
$$\chi_{r,s}^{}(i) := \left\{
\begin{array}{ll}
i\!+\!s & \mbox{if } i \leq r \\
i\!-\!r & \mbox{if } r < i \leq r\!+\!s \\
i   & \mbox{if } i > r\!+\!s \\
\end{array}
\right\} \mbox{ and }
\pi^\#(i) := \left\{
\begin{array}{ll}
\pi(i) & \mbox{if } \pi(i) < \pi(n\!-\!1) \\
\pi(i)\!+\!1 & \mbox{if } \pi(i) > \pi(n\!-\!1) \\
\pi(n\!-\!1)\!+\!1 & \mbox{if } i = n \\
\end{array}
\right\}.$$

\begin{definition}
\label{defrhob}
\rm
For $\pi \in S_n$ with $\pi(1) = 1, \pi \neq {\rm id}_n$ set
$p := \max \{ \;j \;\vert \; \pi(i) = i \mbox { for all } i \leq j\;\}$
and $q := \pi^{-1}(p+1)$. If $\pi(n) \neq 2$ and $\pi(n) \neq n$ let $\pi^\prime \in S_{n-1}$
be the permutation given by
$$\pi^\prime (i) :=\left\{
\begin{array}{ll}
i & \mbox{if } i = 1 \mbox{ or } n-p < i \leq n-1\\
\pi(q+1-i) & \mbox{if } 2 \leq i \leq q-p\\
\pi(i+p) & \mbox{if } q-p < i \leq n-p \\
\end{array}
\right\}.$$
Finally for any $\pi \in S_n$ with $3 \leq n \leq m$ set
$$\rho_{B,m}^{} (\pi) := \left\{
\begin{array}{ll}
(1)(2)(3\;4\;5)+{\rm id}_5-2{\rm id}_4+{\rm id}_3 & \mbox{ if } \pi = {\rm id}_m \mbox{ and } m \geq 5 \\
\pi - \pi\tau_p +(-1)^{q-p}\Upsilon_{n,q-p-1}(\pi^\prime)& \mbox{ if }n = m \mbox{ and } \pi(1)=1 \mbox { and } \pi(n) \neq 2,n\\
\pi - \Upsilon_{n,0}(\pi)& \mbox{ if } 3 \leq n < m \mbox{ and }\pi(1)=1\\
0 & \mbox{ otherwise}
\end{array}
\right.
$$
\end{definition}

\noindent
Now we have our second candidate for an ubr-algorithm $B_m := ({k},\bigcup_{n=3}^{m} S_n^{},\varphi_{B,m}^{},\delta_B^{},\rho_{B,m}^{})$.

\section{Justification of the algorithms A and B}
\label{sectproof}

\begin{theorem}
$A_m$ is an ubr-algorithm for $\P_m$ if $m \geq 2$, $B_m$ is an ubr-algorithm
for $\P_m$ if $m \geq 3$.
\end{theorem}

To prove this, we have to verify the four conditions that we required in section
\ref{sectprin}. The second one is fulfilled by construction, because in the definition of $\delta$
we used reducing moves. There can be at most $\#S-1$ reducing steps for elements
in a linearly ordered, finite set $S$. So $j=\#S-1$ is an integer satisfying the
condition $\delta^j = \delta^{j+1}$.
The remaining parts of the proof of this theorem are given in the rest of this section.

\begin{corollary}
\label{corineq}
We have the inequalities
$$O_A(m)\;\geq\; {\rm rk\ } \P_m \;\geq\; O_C(m)  \;\;\;(\mbox{for }m \geq 2) \mbox{\quad and\quad }
O_B(m) \;\geq\; {\rm rk\ } \P_m \;\;\;(\mbox{for }m \geq 3)$$
where $O_A, O_B, O_C$ are given by $\;O_A(m)\; := \;\mbox{Output}(A_m)$, $\;O_B(m) \;:= \;\mbox{Output}(B_m)$ and
$O_C(m)\; := \;\dim \tilde \Phi_m \big(\Zmodule{\mbox{ caterpillar diagrams of degree } m\; }\big)$.
\end{corollary}

\subsection{Verification of the first condition}

In a first step we show that already the simply connected circle
diagrams generate ${\cal P}$.
One should recall that the circle on which the univalent vertices have been glued
is just a means of visualization, not a part of the diagram.
By a {\it loop} of a diagram we mean a closed path of consecutive edges
that meets each edge at most once.
Obviously, a loop contains only trivalent vertices and it cannot encounter a
vertex twice.
A vertex is called {\it bound} if there exists a loop going through that vertex,
otherwise it is called {\it free}.

We have a threefold partition of circle diagrams, given by their degree,
the dimension of the first homology and the number of free trivalent
vertices: $$D_{m,k,n} := \{\; D \;\vert\; D \mbox{ has }
2m \mbox{ vertices, } \dim H_1(D) = k,\; n \mbox{ of the trivalent vertices are free }\}$$

\begin{lemma}
\label{lemmatrees}
Any element of $D_{m,k,n}$ with $k>0$ and $n>0$ can be expressed (over $\P_m$) in terms
of elements of $D_{m,k,n-1}$.
\end{lemma}
{\bf Proof\quad}
Any $D \in D_{m,k,n}$ has by assumption both bound and free trivalent vertices.
$D$ is connected, so there exists an edge
connecting a free trivalent vertex $f$ with a bound vertex $b$.
Then the application of the IHX relation on this edge
presents $D$ as difference of two diagrams $\in D_{m,k,n-1}$:
\cepsfig{fig8.ps}

\noindent
The IHX does not change the homology, but the diagrams on right side
have one free trivalent vertex less. \qed

\begin{lemma}
\label{lemmatrees2}
Any element of $D_{m,k,0}$ with $k>0$ can be expressed (over $\P_m$) in terms
of elements of $\bigcup_{i} D_{m,k-1,i}$.
\end{lemma}
{\bf Proof\quad}
Any circle diagram has at least one free vertex because, by definition, it
has one or more univalent vertices. For $k > 0$ any $D \in D_{m,k,0}$ has
bound vertices. There must be an edge connecting a free vertex $f$ with
a bound vertex $b$. By assumption $f$ has to be a univalent vertex.
An application of the STU-relation at $f$ opens the loops going through $b$
without introducing new loops:
\cepsfig{fig7.ps}

\noindent
The diagrams $D_1, D_2$ on the right satisfy $\dim H_1 (D_i) = \dim H_1 (D)-1$. \qed

\begin{proposition}
\label{propphia}
$\{D^A_\pi\;\vert\;\pi\in S_{m-1}\}$ generates $\P_m$.
\end{proposition}
{\bf Proof\quad}
Lemmas \ref{lemmatrees} and \ref{lemmatrees2} imply that
the simply connected diagrams $D_{m,0,m-1}$ generate $\P_m$
(simply connected diagrams of degree $m$ must have $m-1$ trivalent vertices).

For $D \in D_{m,0,m-1}$ we choose two neighbouring (with respect to the cyclic ordering)
univalent vertices $a$, $b$. Let $P$ denote the (uniquely determined) path
in $D$ from $a$ to $b$.

If there are trivalent vertices that do not lie on
$P$, we can use exactly the same argument as in lemma \ref{lemmatrees} to
increase the number of vertices on $P$. We finally end up with diagrams
that have a path $P$ going through all trivalent vertices and that connects
two neighbouring univalent vertices.
Because of the AS relation, we can sippose that all $m-1$ edges branching off $P$
lie on the left side of $P$. All these diagrams are of the
form $D^A_\pi$ with $\pi\in S_{m-1}$.\qed

\begin{proposition}
\label{propphib}
$\{D^B_\pi\;\vert\;\pi\in S_m\}$ generates $\P_m$ for $m \geq 3$.
\end{proposition}
{\bf Proof\quad}
We want to show that $D_{m,1,0}$ is a set of generators.
Because of lemma \ref{lemmatrees} and proposition \ref{propphia}, we only have to
show that every simply connected circle diagram can be expressed in terms of
diagrams having one loop, i.e.~elements of $\bigcup_i D_{m,1,i}$.
Let $D \in D_{m,0,m-1}$ with $m \geq 3$, then $D$ has a
trivalent vertex $t$ that is connected to two univalent vertices $a, b$.
We can "throw out" all other univalent vertices between $a$ and $b$ on the
circle, by using STU relations:
\cepsfig{figout.ps}

\noindent
The second diagram on the right has a loop, so it remains to show that
a diagram with a trivalent vertex that is connected to two neighbouring
univalent vertices and another trivalent vertex, is equivalent to a diagram
with a loop. This is done by the following observation:
\cepsfig{fig7_.ps}

\noindent
We have shown that diagrams having one loop and no free trivalent vertices
generate $\P_m$ for $m \geq 3$. These diagrams are equivalent
by the AS relation to $\pm D^B_\pi$ for some $\pi \in S_m$. \qed

\subsection{Verification of the third condition}

\begin{proposition}
The maps $\delta_A$ and $\delta_B$ induce the identity in $\P$.
\end{proposition}
{\bf Proof\quad}
We verify this for every move separately.
\begin{itemize}
\item[I:]
If the left and right side of this move are called $\pi$ and $\pi^\prime$,
then $D^A_\pi$ and $D^A_{\pi^\prime}$ have little triangles on the upper and lower end
(two of the sides are edges of the diagram, the third is part of the circle).
The second picture in the proof of proposition \ref{propphib}
together with the fact that $t \in \Lambda$, implies that
one can push the triangle down through the whole diagram, so $D^A_\pi = D^A_{\pi^\prime}$.
\item[II:] This move is an application of the STU relation; the triangle
in the second term on the right has been pushed down.
\cepsfig{fig11.ps}

\item[III:]
The four moves of type III occur, when the following diagram identities are resolved by
STU on the right and IHX on the left in two different ways that are
indicated by the small arrows.
\cepsfig{fig12.ps}

\item[I$^\prime$:]
The diagrams $D^B_\pi$ are given by putting the permutation $\pi \in S_n$ between
a loop and an oriented circle. The multiplication with $\nu_n$ on the left or
right does not change the diagram, because it just slides vertices on
the loop or on the circle. The multiplication with $\mu_n$ corresponds to
a flip of the loop. When $n$ is odd the AS relation causes the sign to
change during the flip.
\item[II$^\prime$:]
The relation is of the form $\pi \rightarrow \pi_1 - \pi_2$ with
$\pi,\pi_1 \in S_n$ and $\pi_2 \in S_{n-1}$. The picture for type II
makes clear that $D^B_\pi = D^B_{\pi_1} - t\;D^B_{\pi_2}$.
\end{itemize}
Since $\delta_A$ and $\delta_B$ are defined by these moves, we have
hereby shown that $\hat\varphi_A\circ\delta_A = \hat\varphi_A$ and $\hat\varphi_{B,m}\circ\delta_B = \hat\varphi_{B,m}$.
\qed

\subsection{Verification of the forth condition}

First we have to understand $\Theta_k$.
For that purpose, we draw pictures with rectangular boxes named
$\Theta_k$ with $k$ entries on the left and $k$ exits on the right.
In each such box the pictures of all permutations occurring in
$\Theta_k$ shall be inserted (forgetting the upper $n-k$ constant strands)
 and the sum over all resulting diagrams (with the given signs) is taken.
In this way a picture with a $\Theta_k$-box in fact represents a linear
combination of $2^k$ diagrams.

\begin{lemma}
\label{lemmatheta}
In ${\cal A}$ the following relation holds:
\cepsfig{fig13.ps}
\end{lemma}

\noindent
{\bf Proof\quad} (by induction on $k$)

\epsfxsize = 13cm  \epsfig{fig14.ps} \qed

\begin{proposition}
$\hat\varphi_A(\pi) = (-1)^{n-k}\hat\varphi_A(\Theta_k \pi^\prime)$ for any
$\pi \in S_n$.
\end{proposition}
{\bf Proof\quad}
The external vertex on the lower side of $\hat\varphi_A(\pi)$ is named A, the
one on the upper side B and the lowest on the right side C. We will
rotate the circle clockwise, moving C $\rightarrow$ A $\rightarrow$ B.
This operation looks like this (in the picture of $\pi$
the line going from $\pi^{-1}(1)$ to $1$ has been drawn, the other lines have been omitted):
\cepsfig{fig15.ps}

\noindent
To get the second equality, one has to pull straight the path from C to A.
To do this the $n-k = \pi^{-1}(1)-1$ lowest trivalent vertices are swapped, which
is the reason for the factor $(-1)^{n-k}$.
The permutation in the box on the right side is $\pi^\prime$ of definition \ref{defrhoa}.
So by lemma \ref{lemmatheta} the third diagram in the equation is
equivalent to $\hat\varphi(\Theta_k \pi^\prime)$. \qed

\begin{lemma}
\label{lemmups}
For any $\pi\in S_n$, $0 \leq k \leq n-3$ the element $\hat\varphi_{B,n} \big(\Upsilon_{n,k}(\pi)\big)$ of $\P_{n+1}$
is equivalent to the following diagram.
\cepsfig{fig16.ps}
\end{lemma}
{\bf Proof\quad}
The permutation $\pi^\# \in S_{n+1}$, which is used in the definition of $\Upsilon$
in section \ref{sectone}, is obtained
by doubling the $n-th$ string of $\pi$. So the two terms of $(1-\tau_n)\pi^\#$
allow a STU relation, after the right endpoints have been glued to the circle.
Together with our knowledge of $\Theta_k$, we get the following
diagram for $\chi_{1,n-k-2}^{} \; \Theta_{n-k-2}^{} \; \chi_{n-k-2,k+1}^{} \; (1 - \tau_n^{})\; \pi^\#$:
\cepsfig{fig17.ps}

\noindent
This results in the diagram of the claim.\qed

\begin{proposition}
$\hat\varphi_{B,m}^{}\big(\rho_{B,m}^{}(\pi)\big) = 0$ for all $\pi \in S_n$ with $3 \leq n \leq m$.
\end{proposition}
{\bf Proof\quad}
Let $x := (2\;1)(3\;5)(4) + (1)(2\;4)(3)$, then two moves of type I yield
$$\hat\varphi_{B,5}^{}(x) = -\hat\varphi_{B,5}^{}({\rm id}_5) - \hat\varphi_{B,5}({\rm id}_4)$$
Making three moves of type II$^\prime$ we get
$$\hat\varphi_{B,5}^{}(x) = \hat\varphi_{B,5}^{}\big((1)(2)(4)(3\;5)\big) = \hat\varphi_{B,5}^{}\big((1)(2)(3\;4\;5) - {\rm id}_4 + {\rm id}_3\big)$$
Both equations together imply
$\hat\varphi_{B,m}^{}\big((1)(2)(3\;4\;5)+{\rm id}_5-2{\rm id}_4+{\rm id}_3\big) = 0$
for $m \geq 5$.

\hspace{\fill}\\
For any $\pi \in S_m$ and $1 \leq q < m$, the STU relation allows
us to write $\hat\varphi(\pi) - \hat\varphi(\pi\tau_p)$ as a single diagram $D$ with
$\dim H_1(D) = 2$. For the second case in the definition of $\rho_{B,m}^{}$,
we keep the newly introduced loop of $D$ and eliminate the old one.
We show how this is done in the example $m = 9, p = 3, q = 7$:
\cepsfig{fig18.ps}

\noindent
During this operation the diagram $D$ is not changed, except that the
$q-p-1$ trivalent vertices at between $p$ and $q$ are swapped with
the AS relation.
The permutation on right of the dotted line is $\pi^\prime \in S_{m-1}$
given in definition \ref{defrhob}. So by lemma \ref{lemmups}, the last diagram in the equation
is $\Upsilon_{n,q-p-1}(\pi^\prime)$, and we get
$\hat\varphi_{B,m}\big(\pi - \pi\tau_p +(-1)^{q-p}\Upsilon_{n,q-p-1}(\pi^\prime)\big) = 0$.

\hspace{\fill}\\
If $\pi \in S_n$ with $3 \leq n < m$ and $\pi (1)=1$, let $D_0 := D^B_\pi$.
Let the $D_1$ be the result of inserting a triangle at the lowest trivalent vertex of $D_0$.
Then $\varphi(\pi) = t^{m-n}D_0 = t^{m-n-1}D_1$.
Lemma \ref{lemmups} shows that $D_1 = \Upsilon_{n,0}(\pi)$, and so we have
$\hat\varphi_{B,m}\big(\pi - \Upsilon_{n,0}(\pi)\big) = 0$. \qed

\section{Remarks to the implementation}
\label{sectimpl}
\subsection{Dimensions}
One reason for the success of the presented algorithms is that the dimension of ${\cal I}$
($=$ number of irreducible permutations) is surprisingly small.
The moves we are using are very powerful, in the sense that
only a small number of permutation survive the reductions.
We tried out a large number of additional moves, but no considerable
improvement has been achieved this way.

\noindent
The following table displays
the number of permutations and the number of irreducible
permutations, for the ubr-algorithms $A$ and $B$:

\smallskip
\noindent
{
\footnotesize
\begin{tabular}{||c|c|c|c|c|c|c|c|c|c|c|c|c||}
\hline
m                     &$ 2 $&$  3 $&$ 4 $&$  5 $&$   6 $&$   7 $&$    8 $&$     9 $&$     10 $&$      11 $&$       12 $\\
\hline
$\dim        S_A$     &$ 1 $&$  2 $&$ 6 $&$ 24 $&$ 120 $&$ 720 $&$ 5040 $&$ 40320 $&$ 362880 $&$ 3628800 $&$ 39916800 $\\
$\dim {\cal I}_A$    &$ 1 $&$  1 $&$ 2 $&$  5 $&$  16 $&$  64 $&$  301 $&$  1583 $&$   9145 $&$   57449 $&$   389668 $\\
\hline
$\dim        S_B$     &$   $&$  6 $&$ 30 $&$ 150 $&$ 870 $&$ 5910 $&$ 46230 $&$ 409110 $&$ 4037910 $&$ 43954710 $&$ 522956310 $\\
$\dim {\cal I}_B$    &$   $&$  1 $&$ 2 $&$  5 $&$  10 $&$  24 $&$  78 $&$  331 $&$  1685 $&$   9589 $&$   59782 $\\
\hline
\end{tabular}
}
\smallskip

\hspace{\fill}\\
The second reason for the success is that $\rho$ is "complicated enough"
to reproduce the kernel of $\hat\varphi$. It should not surprise that
the "correct" $\rho_{A}^{}$ and $\rho_{B,m}^{}$ have been
found by an intensive trial and error process.
We did not expect that the calculated upper bounds are sharp;
in fact, the algorithms described here are modifications of parts of a
much bigger program that computed the "exact" value ${\rm rk\ } \P$.

\subsection{Hints to the implementations}
\label{hintsec}

At first a list of irreducible permutations for the desired degree $m$ should
be made. Then $\rho(\pi)$ is calculated for any $\pi$ in this list.

The real difficulty is to evaluate $\bar\rho(\pi) = \Delta(\rho(\pi))$.
The simplest idea is to consecutively apply $\delta$, until a linear
combination of irreducible elements is reached. But this would be much too
slow for the interesting degrees $> 9$, because the
reduction trees are too nested.

The solution is to do it upside down. After assigning values to the minimal
permutations, we go from small permutations to the bigger ones. If
we know the values of all permutations smaller than $\pi$, then
the value of $\pi$ is given by a single application of $\delta$ and
picking at most $7$ values out of the table.

One has to assign a $\dim {\cal I}$-dimensional vector to every $\pi$.
A short look at the dimensions shows that keeping this table in memory exceeds
the capacity of any computer. But one can do the calculation component
per component.
Our implementation does $32$ components at a time,
yielding $32$ rows of the matrix for $\bar\rho$ in each run.

Even if each matrix entry uses only one bit, a file in which the matrix
for $\bar\rho_{B,12}^{}$ is saved contains 426 megabytes.
This is one reasons why we are working with $k = {\mathbf{F}}_{2}$.
The other reason is that, some time ago, we found diagrams
$x \in {\cal A}_{10}$ for which we could show $2x = 0$ but not $x = 0$.
To find out, wether $A_{10}$ contains elements of order $2$ or not, was
the main stimulus to make these computer computations.

The program that computes then rank and nullity of the matrices is a standard
Gaussian algorithm, which can of course be implemented very efficiently for
$k = {\mathbf{F}}_{2}$.
By the way, the matrices occurring are not at all sparse: about 40\% of the entries are $1$ and
gzip compresses the files only by factors about 0.9.

It is not a bad idea to add a check sum to each row/column in the data
files, because the probability of making an error in reading/writing a bit to
hard disc might (on some systems) not be far enough away from $1:10^{11}$, which is the approximate number
of bits that have to be read (in our implementation).

\section*{Acknowledgment}
I would like to thank C.-F.~B\"odigheimer, Michael Eisermann, Christoph Lamm
and Jens Lieberum for helpful discussions and reading the manuscript.
I am also thankful to the Studienstiftung for financial support and
the Graduiertenkolleg for providing the computer on
which the computation has been performed.

\section*{Computer files}

At the following locations in the internet,

{\tt http://www.uni-bonn.de/\~{}jk/pvi12.html}

{\tt ftp://ftp.uni-bonn.de/usr/jk/pvi12}

\noindent
you will find:
\begin{itemize}
\item
C/Pascal implementations of the algorithms.
\item
a basis for $\P_{m\leq 12}$ that is produced by the algorithm $B$.
\item
the values of $\tilde\Phi$ of the 175 diagrams that are
mentioned in the proof of theorem \ref{theop}.
\item
a file containing this paper (or a newer version of it).
\item
a summary of the results of section \ref{sectres} and possibly some newer data.
\end{itemize}

redpajama_set_name: RedPajamaArXiv


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Příjmení Rimmer nesou nebo nesly tyto žijící nebo fiktivní osoby:

 Arnold Rimmer – fiktivní postava ze sitcomu BBC Červený trpaslík
 Davey Rimmer (* 1968) – britský baskytarista
 David Rimmer (* 1942) – kanadský filmový režisér
 Eve Rimmer (1937–1996) – novozélandská atletka
 Jimmy Rimmer (* 1948) – britský fotbalový brankář
 John Rimmer (1878–1962) – britský atlet, olympionik
 Michael Rimmer (* 1986) – britský atlet, běžec na 800 m
 Michael Rimmer – fiktivní záporný hrdina filmu The Rise and Rise of Michael Rimmer
 Richard Rimmer – ředitel California Department of Corrections and Rehabilitation
 Robert Rimmer (1917–2001) – americký spisovatel, autor knihy The Harrad Experiment
 Shane Rimmer (1936–2019) – kanadský herec
 William Rimmer (1816–1879) – americký malíř a sochař
 William Rimmer (1862–1936) – britský hudební skladatel

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