Abstract. There is compelling evidence including some long-route indirect proofs, that the following are the same, or at least closely and directly related to each other:
There is also some weaker evidence that all these topics remain related even when the various commutativity, co-commutativity and other restrictions are dropped.
- The Kashiwara-Vergne conjecture, which says, more or less, that convolutions of Ad-invariant functions on a Lie group are the same as convolutions of Ad-invariant functions on its Lie algebra.
- Drinfel'd associators, at least "restricted to one loop", and the work of Alekseev and Torossian.
- Knot invariants, and in particular, the multi-variable Alexander polynomial.
- Virtual knots in which overcrossings (but not undercrossings) are made to commute.
- The Etingof-Kazhdan formalism in the case of co-commutative Lie bi-algebras.
If all goes well, in 5-10 years I will be able to give a coherent explanation of all that (perhaps sooner, if others will get interested and do the work). For now, all I have are some disorganized tidbits.
Partially I will follow the topics and handouts of several talks that I gave recently. In particular, with base:=http://www.math.toronto.edu/~drorbn/Talks, see base/Sandbjerg-0810, base/Istanbul-0606, base/Zurich-080513, base/MSRI-0808, and base/Copenhagen-081009.
Today's Handout. UVWKnots.html, UVWKnots.pdf.
The Key Points.