Knot diagrams, sets of moves and equivalence relations

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Short version: Does anyone study equivalence classes generated by a given set of "moves" (in the sense of, but not limited to, Reidemeister moves) on the set of knot diagrams?

Yes, I understand that the concept of knot has a natural geometrical significance, that one usually views knot diagrams as a tool to study underlying knots, that the value of Reidemeister moves lies in how they preserve and generate the equivalence relation of isotopy. So, yes, I see why a knot theorist might reasonably have little interest, say, in looking at local moves not preserving isotopy.

So I'm asking this question in the spirit of abstraction for its own sake. But there is a precedent. A "symmetry theorist" studies groups because they capture the set of symmetries of important objects. But combinatorial group theory studies equivalence classes of strings under moves...and symmetry, if it enters the story at all, does so as a tool.

That said, it would be interesting if one could add an extra move to the Reidemeister moves that produced a coarser but computationally tractable classification.

No need to retread the ground here -- http://en.wikipedia.org/wiki/Reidemeister_move -- so, for example, I already understand that knot theorists know what happens with only Reidemeister type II and III moves. I am interested in stories like this, where one gets a finer equivalence relation that isotopy, but equally interested in sets of local moves that don't preserve isotopy and thus generate equivalence relations either coarser to, or simply incomparable with, isotopy.

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The appropriate abstraction here seems to me to be a category presented by generators and relations. (The category relevant to knots is the tangle category: math.ucr.edu/home/baez/tangles.html) This is a very general construction: it has as special cases monoids presented by generators and relations, as well as posets... – Qiaochu Yuan Aug 7 at 3:02

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There are lots of such moves. To give an example I am very familiar with, $C_k$-moves are certain moves (surgery along claspers) that generate the equivalence relation that two knots share Vassiliev invariants of order up to k-1. Another example: Lou Kauffman first observed that two knots have the same Arf invariant iff they differ by a sequence of "band-pass" moves. – Jim Conant Aug 7 at 3:13
Kawauchi's survey of knot theory book mentions several of the theorems Jim Conant alludes to. – Ryan Budney Aug 7 at 5:44
@Qiaochu Yuan So a knot or link then belongs to Hom(0,0)? But can't a "move" fail to respect the imposed directionality? Can't it wind back and forth in "time," using perhaps some but not all of the strands at a given "time" and thus escape this optic? – David Feldman Aug 7 at 8:09

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David Feldman and Qiaochu: I think you're both right. The answer is 2-dimensional algebra, which does not suffer from imposed directionality. See the work of Dror Bar-Natan, on "the circuit algebra of tangles" and related 2-dimensional algebras. This is indeed what is happening- the abstraction is to certain diagrammatic algebras (over a "modular operad") in the sense Qiaochu is refering to. And I believe this (intentionally leaving what I mean by "this" slightly vague) is very much the appropriate abstraction. – Daniel Moskovich Aug 7 at 13:07
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4 Answers

Very much so. There are a number of small industries centred around studying equivalence classes of knot diagrams generated by a set of moves.

1. The study of claspers. For example, $C_k$-moves are a special type of clasper surgeries. MathSciNet indicates 123 citations for Habiro's fundamental paper Claspers and finite type invariants of links, providing some coarse measure of the vitality of the topic.
2. Replacing one rational tangle in a knot diagram by another generates an equivalence relation which has been deeply studied using quandles. See e.g. J. Przytycki's introductory lectures.
3. Dehn surgery, where the surgery curve is required to belong to some specified part of a knot group or link group (in the kernel of its representation to some fixed group, for instance) generates equivalence relations on knot diagrams modulo combinatorial "twisting" moves, which have been studied by Cochran-Orr-Gerges, and (excuse the self promotion) by myself and Andrew Kricker, and by Litherland and Wallace. The techniques for studying these equivalence relations have been topological rather than combinatorial.
4. There are a number of setting in which one allows Reidemeister moves plus some crossing changes, but not others. In the theory of finite type invariant, one fixes a some crossings (considers them in resolutions of "double points"), and allows crossing changes away from them. The equivalence classes are detected by the finite-type invariant of type the number of "fixed" crossings. In a similar-sounding vein, a free virtual knot is a virtual knot where we allow crossing changes away from virtual crossings. They have a rich theory- see e.g. this Manturov paper.
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The Delta move generates the equivalence relation of linking numbers for links, as proved by Matveev and Murakami Nakanishi.

There's the double Delta move which Naik and Stanford showed generate S-equivalence.

A student of Freedman studied "slide equivalence" of knot diagrams. Unfortunately it wasn't published though.

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These both delta moves are special cases of Y-clasper surgeries ($C_2$ moves). Take a clasper with one leaf ringing around each strand participating in the delta. Could you describe slide equivalence? – Daniel Moskovich Aug 7 at 15:57

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@Daniel: I see, I learned about these moves in grad school I think before claspers were around, and I didn't realize they were equivalent. The number of Delta moves needed to unknot a knot is congruent to the Arf invariant. I don't have a copy of Huang's thesis, but I went to his thesis defense, and from what I can remember, you can slide arcs of the knot diagram around, treating a strand going under a crossing as two separate arcs attaching at the overstrand. But this was over 15 years ago, so I don't remember the details. – Agol Aug 7 at 16:54

Legendrian knots

Legendrian knots are smooth knots whose tangent directions are contained in a contact structure such as the standard contact structure on $\mathbb{R}^3$, $dz=y~dx$. Every knot has Legendrian representatives. Two Legendrian knots of the same topological type might not be isotopic through Legendrian knots.

The projection of a Legendrian knot in the standard contact structure to the $xz$-plane is called its front projection. Some people study Legendrian knots through diagrams showing front projections. The $y$ coordinate can be recovered from the slope, so all crossings are determined by the diagram. However, there can be no vertical tangencies, since $y$ would be undefined, and you must allow cusps. See this Notices article.

There are analogues of Reidemeister moves, so this gives a refinement of knot theory described by a set of diagram moves on front projections.

Actually, you don't have to work with cusped diagrams. You can make all cusps horizontal, and you could choose to replace the horizontal cusps with vertical tangencies. So, standard knot diagrams up to a restricted set of moves (including disallowing some isotopies where no Reidemeister move was performed, but where vertical tangencies would have been introduced or removed) are equivalent to Legendrian knots.

Suppose you study curves up to isotopy instead of the standard ambient isotopy used in knot theory. You may be disappointed: knot theory in $S^3$ becomes trivial. You are allowed to replace a piece of a diagram showing a long knot with a long unknot by shrinking the knot to a point and forgetting it. However, link theory is still nontrivial, and so is knot theory in a $3$-manifold which is not simply connected. See Rolfsen, "Localized Alexander Invariants and Isotopy of Links." Annals of Mathematics 101 (1975) 1-19.

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One classical example of such move is Conway mutation, which falls into the category of tangle replacement, as Qiaochu Yuan mentioned in his comment. There's a very famous pair of mutants, the Kinoshita-Terasaka and the Conway knot (see the wikipedia article).

Apparently, there's some topology behind this move: recently, using knot Floer homology, Josh Greene has shown that two alternating knots are mutants if their branched double covers are homeomorphic, and the other arrow was shown by Viro (see references in Greene's paper).

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That link leads to an empty page. – Will Sawin Aug 8 at 0:15
Oops, sorry. I've fixed it now. Thanks. – Marco Golla Aug 8 at 6:57
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