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Virtual knot theory is an interesting generalization of knot theory in which ``virtual" crossings are allowed. See Kauffman's Virtual Knot Theory for an introduction. Greg Kuperberg gave a nice topological interpretation of virtual knots in this paper. One reason to be interested in virtual knots and links is that many knot and link invariants generalize to the virtual setting. For example, my naive understanding is that virtual knots are a more natural domain for Vassiliev invariants than knots are. My question is whether anyone knows of examples that demonstrate the utility of virtual knot theory? For example, are there any interesting theorems outside of virtual knot theory that can be most easily proven using virtual knot theory? The papers I have seen seem to pursue VKT for its own sake, but my sense is that such a natural area must be of much wider use.

Added: I just ran across a paper by Rourke that explains a geometric interpretation of "welded links" which are like virtual links but an additional move is allowed. This is a beautiful little paper which explains how welded knot theory corresponds to a theory of certain embedded tori in $\mathbb R^4$. It's really amazing how the Reidemeister moves, both virtual and classical, correspond to isotopies of what Rourke calls toric links. This is a part of Bar-Natan's program mentioned by Theo Johnson-Freyd and Daniel Moskovich below. Dror calls the toric links "flying rings".

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In that article you cite, Kaufmann mentions the thickened surface motivation above, plus a combinatorial motivation. I suppose knots in thickened surfaces you could think of as a "generic" picture of a knot in a 3-manifold equipped with a Heegaard splitting. But it's not clear to me this is anything more than a fairly technical observation -- I mean, I don't know if it goes anywhere interesting. –  Ryan Budney Dec 1 '10 at 16:11

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Here are two ways to think of knots:

  1. As ambient isotopy clases of smooth embeddings of S1 in S3.
  2. As a planar algebra generated by over-crossings and under-crossings, modulo Reidemeister moves.

Quantum topology makes ample use of the second viewpoint. But if you're viewing a knot as an element of a planar algebra, or, well, as an operad, then the more natural operad to work with would be one in which endpoints of crossings get matched up- abstractly, in a graph theoretical sense, rather than by lines in a plane. Bar-Natan calls such a structure a "circuit algebra" (a modular operad?). In quantum topology, you're looking for a homomorphic expansion of such an operad to some Lie-algebraic object, which carries a parallel operadic structure, such as for example the Drinfeld double of a finite group. The point now is that a circuit algebra is algebraically better behaved than a planar algebra, and so it's easier to find homomorphic expansions and to calculate them- and they tell you something about Lie bialgebras. In particular, homomorphic expansions of virtual knots knotted trivalent graphs should tell you about Etingoff-Kazhdan quantization of Lie bialgebras.
Knots are more complicated, because the planarity restriction for such operadic structures interacts badly with the Lie algebraic structure, and you end up with associators. Indeed, specifying a homomorphic expansion for knots (or for KTG's, to be more precise) is the same thing as specifying a (nice) associator. Nobody quite knows how to handle associators. Therefore it's algebraically sensible to pass to circuit algebras, where you obtain invariants which respect the operadic structure (virtual knot invariants extend to virtual tangles- but knot invariants only extend to "non-associative tangles" or to "q-tangles"- not to tangles- because of the presence of associators).

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I like the idea that virtual knot invariants extend to virtual tangles, as opposed to q-tangles. –  Jim Conant Dec 2 '10 at 7:58

I highly recommend that you check out some of the recent work by Dror Bar-Natan. In particular, he videotapes and posts online all his talks: http://www.math.toronto.edu/drorbn/Talks/

In some of his recent talks, he explains various connections between virtual knots and questions in knot theory but also in representation theory and Lie theory.


I (Theo, who posted this answer originally) have set the answer to CW after it was pointed out that I really should have written more. Rather than me writing more, I'd like to invite others to elaborate if they'd like (I'm not an expert). To start it off, the following paragraph is precisely a comment by Daniel Moskovich.

Dror's programme, in some sense, is to turn quantum topology on its head by making algebra (Lie algebras in particular) the primary object, and topological knotted stuff the tool used to understand them. Specifying a universal finite-type invariant for KTG's is the same thing as specifying a (nice) associator. In the same sense, specifying a universal finite-type invariant for v-KTG's should really explain Etingoff-Kazhdan quantization of Lie bialgebras. This is discussed in his Montpellier talk. It's an ambitious programme, but one with every reason to succeed, IMHO.

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Neat. I never noticed that section of Dror's website before! Do you happen to know which talks mention virtual knot theory? –  Jim Conant Dec 1 '10 at 17:47
    
@Theo: could you be a little more precise? –  Ryan Budney Dec 1 '10 at 17:55
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Dror's programme, in some sense, is to turn quantum topology on its head by making algebra (Lie algebras in particular) the primary object, and topological knotted stuff the tool used to understand them. Specifying a universal finite-type invariant for KTG's is the same thing as specifying a (nice) associator. In the same sense, specifying a universal finite-type invariant for v-KTG's should really explain Etingoff-Kazhdan quantization of Lie bialgebras. This is discussed in his Montpellier talk. It's an ambitious programme, but one with every reason to succeed, IMHO. –  Daniel Moskovich Dec 1 '10 at 19:51
    
@Ryan: Better, I've switched the answer to CW, so that many people should feel free to add comments on Dror's work on virtual knots. In particular, I think that @Daniel Moskovich's comment is very good, so I've incorporated it into the answer. –  Theo Johnson-Freyd Dec 2 '10 at 3:17

My view of virtual knot theory: An R-matrix is a tensor with four indices that, in a natural sense, satisfies the Reidmeister relations. Actually it is enough to consider the third Reidemeister relation, and assume that the R-matrix has an inverse. This definition is motivated by the problem of constructing link invariants: Any link can then be read as a word in R matrices, and the value of this word for a given R-matrix is a link invariant. But now you could ask, what are all possible words in an R-matrix, and when are two words symbolically equivalent? (This is a possibly stricter condition than being equal for a particular R-matrix or even all R-matrices.) It's clear that if two words come from two presentations of a link, then they are symbolically equivalent. However, there are also words that don't come from links. By definition, an arbitrary R-matrix word is a virtual link or, if it has free indices, a virtual tangle.

So the question of classifying virtual links or tangles is the "word problem" for R-matrices. My theorem gives a solution to the word problem. I show that every virtual link has a unique minimum-genus representation as a link in a thickened surface. The interpretation as a link in a surface is not really due to me. Rather, the new result is the uniqueness theorem. It yields an algorithm to determine when R-matrix words are symbolically equivalent.

But, as with word problems for groups, you can always imagine an R-matrix that satisfies extra relations, and ask the question again. One interesting case is Louis Kauffman's "virtualization" move. My solution doesn't work with this extra move. It is an interesting open problem, again due to Lou, whether there are two inequivalent links that are equivalent as virtual links together with the virtualization move.

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This is a good answer, but it's worth noting that your theorem relies on the Reidemeister I move being true locally. This relation does not hold for the R-matrices from quantum groups, so you don't actually get invariants of virtual knots in the way you sketch in your answer, but rather virtual knots up to isotopy. I'm not aware of a uniqueness or classification result in this context. –  Dylan Thurston Jun 7 '12 at 0:28

This paper by Chrisman and Manturov is as close as possible to an answer to my original question. From their introduction:

By classical knot theory we mean the study of knots and links in the $3$-sphere. By virtual knot theory we mean the study knots and and links in thickened surfaces $\Sigma\times I$ modulo stabilization, where $\Sigma$ is compact orientable surface (not necessarily closed), and I is the closed unit interval. The goal of the present paper is to study classical knots using the methods of virtual knot theory. To do this, we introduce the concept of a virtual cover of a classical knot.

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