The Envelope of The Alexander Polynomial
The information below is preliminary and cannot be trusted! (v)
The Internal Kernel of the Alexander Polynomial
All that there is here comes from 06-1350/Class Notes for Tuesday October 24. Many further relevant facts are in arXiv:q-alg/9602014 by José M. Figueroa-O'Farrill, Takashi Kimura, Arkady Vaintrob and in arXiv:math.QA/0204346 by Jens Lieberum.
At the moment I know of just three relations in the internal kernel of the Alexander polynomial: the bubble relation, the H relation and the 4Y relation:
I have good reasons to suspect that there are further relations. But at the moment I don't know what they are, so below we will make do with what we have.
The VS-Algebra Envelope of the Alexander Polynomial
Let denote the Conway polynomial and denote the Alexander polynomial. By [Bar-Natan_Garoufalidis_96] we know that
is a canonical Vassiliev power series. Let d denote "half a bubble". The following theorem follows easily from the above canonicity statement and the fact that (in shorter and less precise form, ), where is the weight system of the Alexander-Conway polynomial:
Theorem. Let be a knot and let be the Kontsevich integral of . Then within the envelope of the Alexander-Conway polynomial,