06-1350/Class Notes for Thursday October 12 - Revision history

Andy: Added scanned class notes.

2006-10-24T20:28:47Z

Added scanned class notes.

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	{{note\|Zagier_01}} D. Zagier, ''Vassiliev invariants and a strange identity related to the Dedekind eta-function,'' Topology '''40(5)''' (2001) 945-960.		{{note\|Zagier_01}} D. Zagier, ''Vassiliev invariants and a strange identity related to the Dedekind eta-function,'' Topology '''40(5)''' (2001) 945-960.

			==Scanned Notes==

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Drorbn at 13:54, 12 October 2006

2006-10-12T13:54:48Z

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	<center><math>D=d\left(1-H\frac{\partial}{\partial H}\right)+H\frac{\partial}{\partial l},</math></center>		<center><math>D=d\left(1-H\frac{\partial}{\partial H}\right)+H\frac{\partial}{\partial l},</math></center>
	acting on polynomials in the variables <math>d</math>, <math>H</math> and <math>l</math>. More precisely - determine the quotient of the space of polynomials in <math>d</math>, <math>H</math> and <math>l</math> by the image of <math>D</math>. More generally, it is a standard subject in algebra to deal with the images and cokernels of algebraic operators; the results are ideals and the quotients are often-studied algebras. But where's the theory for the images and "cokernels" of differential operators.		acting on polynomials in the variables <math>d</math>, <math>H</math> and <math>l</math>. More precisely - determine the quotient of the space of polynomials in <math>d</math>, <math>H</math> and <math>l</math> by the image of <math>D</math>. More generally, it is a standard subject in algebra to deal with the images and cokernels of algebraic operators; the results are ideals and the quotients are often-studied algebras. But where's the theory for the images and "cokernels" of differential operators.

			===Why do I care?===

			The answer to this question is directly related to the determination of the "envelope" of the Alexander polynomial for two component links.

	==Some Computations==		==Some Computations==

Drorbn at 13:32, 12 October 2006

2006-10-12T13:32:08Z

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			{{06-1350/Navigation}}

	==A Question==		==A Question==

Drorbn at 13:31, 12 October 2006

2006-10-12T13:31:33Z

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			==A Question==

			Determine the image of the differential operator
			<center><math>D=d\left(1-H\frac{\partial}{\partial H}\right)+H\frac{\partial}{\partial l},</math></center>
			acting on polynomials in the variables <math>d</math>, <math>H</math> and <math>l</math>. More precisely - determine the quotient of the space of polynomials in <math>d</math>, <math>H</math> and <math>l</math> by the image of <math>D</math>. More generally, it is a standard subject in algebra to deal with the images and cokernels of algebraic operators; the results are ideals and the quotients are often-studied algebras. But where's the theory for the images and "cokernels" of differential operators.

	==Some Computations==		==Some Computations==

Drorbn at 14:40, 10 October 2006

2006-10-10T14:40:00Z

New page

==Some Computations==

The following table (taken from {{ref|Bar-Natan_95}} and {{ref|Kneissler_97}}) shows the number of type <math>m</math> invariants of knots and framed knots modulo type <math>m-1</math> invariants (<math>\dim{\mathcal A}_m^r</math> and <math>\dim{\mathcal A}_m</math>) and the number of multiplicative generators of the algebra <math>{\mathcal A}</math> in degree <math>m</math> (<math>\dim{\mathcal P}_m</math>) for <math>m\leq 12</math>. Some further tabulated results are in {{ref|Bar-Natan_96}}.

{|align=center border=1 cellspacing=0 cellpadding=4
|- align=right
|align=center|m
|0
|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
|11
|12
|- align=right
|align=center|<math>\dim{\mathcal A}_m^r</math>
|1
|0
|1
|1
|3
|4
|9
|14
|27
|44
|80
|132
|232
|- align=right
|align=center|<math>\dim{\mathcal A}_m</math>
|1
|1
|2
|3
|6
|10
|19
|33
|60
|104
|184
|316
|548
|- align=right
|align=center|<math>\dim{\mathcal P}_m</math>
|0
|1
|1
|1
|2
|3
|5
|8
|12
|18
|27
|39
|55
|}

==Some Further Computations==

Some further computations for links and tangles were made by Siddarth Sankaran and Zavosh Amir-Khosravi. See [[VasCalc Results - ChordMod4T]].

==Some Bounds==

Little is known about these dimensions for large <math>m</math>. There is an explicit conjecture
in {{ref|Broadhurst_97}} but no progress has been made in the direction of proving or disproving it. The best asymptotic bounds available are:

'''Theorem.''' For large <math>m</math>, <math>\dim{\mathcal P}_m>e^{c\sqrt m}</math> (for any fixed <math>c<\pi\sqrt{\frac23}</math>, see {{ref|Dasbach_00}}, {{ref|Kontsevich}}) and <math>\dim{\mathcal A}_m<6^mm!\sqrt{m}/\pi^{2m}</math> ({{ref|Stoimenow_98}}, {{ref|Zagier_01}}).

==Some References==

{{note|Bar-Natan_95}} D. Bar-Natan, ''On the Vassiliev knot invariants,'' Topology '''34''' (1995) 423-472.

{{note|Bar-Natan_96}} D. Bar-Natan, ''Some computations related to Vassiliev invariants,'' electronic publication, <tt>http://www.math.toronto.edu/~drorbn/LOP.html#Computations</tt>.

{{note|Broadhurst_97}} D. J. Broadhurst, ''Conjectured enumeration of Vassiliev invariants,'' preprint, September 1997, {{arXiv|q-alg/9709031}}.

{{note|Dasbach_00}} O. T. Dasbach, ''On the combinatorial structure of primitive Vassiliev invariants III - a lower bound,'' Comm. in Cont. Math. '''2-4''' (2000) 579-590, {{arXiv|math.GT/9806086}}.

{{note|Kneissler_97}} J. A. Kneissler, ''The number of primitive Vassiliev invariants up to degree twelve,'' preprint, June 1997, {{arXiv|q-alg/9706022}}.

{{note|Kontsevich}} M. Kontsevich, unpublished.

{{note|Stoimenow_98}} A. Stoimenow, ''Enumeration of chord diagrams and an upper bound for Vassiliev invariants,'' Jour. of Knot Theory and its Ramifications '''7(1)''' (1998) 94-114.

{{note|Zagier_01}} D. Zagier, ''Vassiliev invariants and a strange identity related to the Dedekind eta-function,'' Topology '''40(5)''' (2001) 945-960.