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".
Best Answer
Here are two ways to think of knots:
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
knotsknotted 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).