Difference between revisions of "The Envelope of The Alexander Polynomial"

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(Alexander-Conway, Precisely)
 
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{{Paperlets Navigation}}
 
{{Paperlets Navigation}}
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{{In Preparation}}
  
 
==The Internal Kernel of the Alexander Polynomial==
 
==The Internal Kernel of the Alexander Polynomial==
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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.
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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:
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[[Image:The Bubble Relation.svg|thumb|330px|center|The Bubble Relation]]
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[[Image:TheHRelation.svg|thumb|550px|center|The H Relation]]
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[[Image:The 4Y Relation.svg|thumb|440px|center|The 4Y Relation]]
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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==
 
==The VS-Algebra Envelope of the Alexander Polynomial==
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==Alexander-Conway, Precisely==
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Let <math>C(z)</math> denote the Conway polynomial and <math>A(t)</math> denote the Alexander polynomial. By {{ref|Bar-Natan_Garoufalidis_96}} we know that
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{{Equation*|<math>\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>}}
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is a canonical Vassiliev power series. Let <math>d</math> denote "half a bubble". The following theorem follows easily from the above canonicity statement and the fact that <math>W_C(d^{2n})=(-2\hbar^2)^n</math> (in shorter and less precise form, <math>\hbar=W_C(id/\sqrt2)</math>), where <math>W_C</math> is the weight system of the Alexander-Conway polynomial:
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'''Theorem.''' Let <math>K</math> be a knot and let <math>Z(K)</math> be the Kontsevich integral of <math>K</math>. Then within the envelope of the Alexander-Conway polynomial,
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{{Equation*|<math>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>.}}
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==References==
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{{note|Bar-Natan_Garoufalidis_96}} Dror Bar-Natan and Stavros Garoufalidis, ''[http://www.math.toronto.edu/~drorbn/LOP.html#MMR On the Melvin-Morton-Rozansky Conjecture],'' Inventiones Mathematicae '''125''' (1996) 103-133.

Latest revision as of 08:13, 1 May 2007

In Preparation

The information below is preliminary and cannot be trusted! (v)

Contents

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.

The VS-Algebra Envelope of the Alexander Polynomial

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/\sqrt2)), 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/\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}).

References

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