Here are two ways to think of knots:
- As ambient isotopy clases of smooth embeddings of S1 in S3.
- 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).
