The Envelope of The Alexander Polynomial

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In Preparation

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:

The Bubble Relation

The H Relation

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.

Alexander-Conway, Precisely

Let $C(z)$ denote the Conway polynomial and $A(t)$ denote the Alexander polynomial. By [Bar-Natan_Garoufalidis_96] we know that

${\frac {\hbar }{e^{\hbar /2}-e^{-\hbar /2}}}C(e^{\hbar /2}-e^{-\hbar /2})={\frac {\hbar }{e^{\hbar /2}-e^{-\hbar /2}}}A(e^{\hbar })$

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 $W_{C}(d^{2n})=(-2\hbar ^{2})^{n}$ (in shorter and less precise form, $\hbar =W_{C}(id/{\sqrt {2}})$), where $W_{C}$ is the weight system of the Alexander-Conway polynomial:

Theorem. Let $K$ be a knot and let $Z(K)$ be the Kontsevich integral of $K$. Then within the envelope of the Alexander-Conway polynomial,

$Z(K)={\frac {id/{\sqrt {2}}}{e^{id/2{\sqrt {2}}}-e^{-id/2{\sqrt {2}}}}}C(e^{id/2{\sqrt {2}}}-e^{-id/2{\sqrt {2}}})={\frac {id/{\sqrt {2}}}{e^{id/2{\sqrt {2}}}-e^{-id/2{\sqrt {2}}}}}A(e^{id/{\sqrt {2}}})$.

References

[Bar-Natan_Garoufalidis_96] ^  Dror Bar-Natan and Stavros Garoufalidis, On the Melvin-Morton-Rozansky Conjecture, Inventiones Mathematicae 125 (1996) 103-133.