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 [math]\displaystyle{ C(z) }[/math] denote the Conway polynomial and [math]\displaystyle{ A(t) }[/math] denote the Alexander polynomial. By [Bar-Natan_Garoufalidis_96] we know that

[math]\displaystyle{ \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) }[/math]

is a canonical Vassiliev power series. Let [math]\displaystyle{ d }[/math] denote "half a bubble". The following theorem follows easily from the above canonicity statement and the fact that [math]\displaystyle{ W_C(d^{2n})=(-2\hbar^2)^n }[/math] (in shorter and less precise form, [math]\displaystyle{ \hbar=W_C(id/\sqrt2) }[/math]), where [math]\displaystyle{ W_C }[/math] is the weight system of the Alexander-Conway polynomial:

Theorem. Let [math]\displaystyle{ K }[/math] be a knot and let [math]\displaystyle{ Z(K) }[/math] be the Kontsevich integral of [math]\displaystyle{ K }[/math]. Then within the envelope of the Alexander-Conway polynomial,

[math]\displaystyle{ Z(K) = \frac{id/\sqrt2}{e^{id/2\sqrt2}-e^{-id/2\sqrt2}} C(e^{id/2\sqrt2}-e^{-id/2\sqrt2}) = \frac{id/\sqrt2}{e^{id/2\sqrt2}-e^{-id/2\sqrt2}} A(e^{id/\sqrt2}) }[/math].

References

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