Difference between revisions of "The Envelope of The Alexander Polynomial"
(→AlexanderConway, Precisely) 

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{{Paperlets Navigation}}  {{Paperlets Navigation}}  
+  {{In Preparation}}  
==The Internal Kernel of the Alexander Polynomial==  ==The Internal Kernel of the Alexander Polynomial==  
+  
+  All that there is here comes from [[061350/Class Notes for Tuesday October 24]]. Many further relevant facts are in {{arXivqalg/9602014}} by José M. FigueroaO'Farrill, Takashi Kimura, Arkady Vaintrob and in {{arXivmath.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:The Bubble Relation.svgthumb330pxcenterThe Bubble Relation]]  
+  
+  [[Image:TheHRelation.svgthumb550pxcenterThe H Relation]]  
+  
+  [[Image:The 4Y Relation.svgthumb440pxcenterThe 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 VSAlgebra Envelope of the Alexander Polynomial==  ==The VSAlgebra Envelope of the Alexander Polynomial==  
+  
+  ==AlexanderConway, Precisely==  
+  
+  Let <math>C(z)</math> denote the Conway polynomial and <math>A(t)</math> denote the Alexander polynomial. By {{refBarNatan_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 <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 AlexanderConway 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 AlexanderConway 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==  
+  
+  {{noteBarNatan_Garoufalidis_96}} Dror BarNatan and Stavros Garoufalidis, ''[http://www.math.toronto.edu/~drorbn/LOP.html#MMR On the MelvinMortonRozansky Conjecture],'' Inventiones Mathematicae '''125''' (1996) 103133. 
Latest revision as of 08:13, 1 May 2007

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 061350/Class Notes for Tuesday October 24. Many further relevant facts are in arXiv:qalg/9602014 by José M. FigueroaO'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 VSAlgebra Envelope of the Alexander Polynomial
AlexanderConway, Precisely
Let denote the Conway polynomial and denote the Alexander polynomial. By [BarNatan_Garoufalidis_96] we know that
is a canonical Vassiliev power series. Let 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 AlexanderConway polynomial:
Theorem. Let be a knot and let be the Kontsevich integral of . Then within the envelope of the AlexanderConway polynomial,
References
[BarNatan_Garoufalidis_96] ^ Dror BarNatan and Stavros Garoufalidis, On the MelvinMortonRozansky Conjecture, Inventiones Mathematicae 125 (1996) 103133.