The Envelope of The Alexander Polynomial - Revision history

Drorbn: /* Alexander-Conway, Precisely */

2007-05-01T13:13:36Z

Alexander-Conway, Precisely

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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>}}		{{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>}}

	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 <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:		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:

	'''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,		'''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,

2007-05-01T01:20:17Z

← Older revision		Revision as of 21:20, 30 April 2007
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	==The VS-Algebra Envelope of the Alexander Polynomial==		==The VS-Algebra Envelope of the Alexander Polynomial==

			==Alexander-Conway, Precisely==

			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

			{{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>}}

			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 <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:

			'''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,
			{{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>.}}

			==References==

			{{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.

2007-04-02T17:55:12Z

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	{{Paperlets Navigation}}		{{Paperlets Navigation}}
			{{In Preparation}}

	==The Internal Kernel of the Alexander Polynomial==		==The Internal Kernel of the Alexander Polynomial==
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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:		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:

			[[Image:The Bubble Relation.svg\|thumb\|330px\|center\|The Bubble Relation]]

	[[Image:TheHRelation.svg\|thumb\|550px\|center\|The H Relation]]		[[Image:TheHRelation.svg\|thumb\|550px\|center\|The H Relation]]

2007-04-01T16:31:17Z

The Internal Kernel of the Alexander Polynomial

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	[[Image:TheHRelation.svg\|thumb\|550px\|center\|The H Relation]]		[[Image:TheHRelation.svg\|thumb\|550px\|center\|The H Relation]]

	[[Image:~~The4YRelation~~.svg\|thumb\|440px\|center\|The 4Y Relation]]		[[Image:The 4Y Relation.svg\|thumb\|440px\|center\|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.		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.

2007-04-01T14:21:11Z

← Older revision		Revision as of 10:21, 1 April 2007
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	==The Internal Kernel of the Alexander Polynomial==		==The Internal Kernel of the Alexander Polynomial==

	~~At the moment all~~ that there is here is at [[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.		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:

			[[Image:TheHRelation.svg\|thumb\|550px\|center\|The H Relation]]

			[[Image:The4YRelation.svg\|thumb\|440px\|center\|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==		==The VS-Algebra Envelope of the Alexander Polynomial==

2006-10-24T02:21:24Z

← Older revision		Revision as of 22:21, 23 October 2006
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	==The Internal Kernel of the Alexander Polynomial==		==The Internal Kernel of the Alexander Polynomial==

			At the moment all that there is here is at [[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.

	==The VS-Algebra Envelope of the Alexander Polynomial==		==The VS-Algebra Envelope of the Alexander Polynomial==

2006-10-23T18:11:41Z

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==The Internal Kernel of the Alexander Polynomial==

==The VS-Algebra Envelope of the Alexander Polynomial==