06-1350/Class Notes for Thursday October 12
Determine the image of the differential operator
acting on polynomials in the variables , and . More precisely - determine the quotient of the space of polynomials in , and by the image of . 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.
The following table (taken from [Bar-Natan_95] and [Kneissler_97]) shows the number of type invariants of knots and framed knots modulo type invariants ( and ) and the number of multiplicative generators of the algebra in degree () for . Some further tabulated results are in [Bar-Natan_96].
Some Further Computations
Some further computations for links and tangles were made by Siddarth Sankaran and Zavosh Amir-Khosravi. See VasCalc Results - ChordMod4T.
Little is known about these dimensions for large . There is an explicit conjecture in [Broadhurst_97] but no progress has been made in the direction of proving or disproving it. The best asymptotic bounds available are:
[Bar-Natan_95] ^ D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423-472.
[Bar-Natan_96] ^ D. Bar-Natan, Some computations related to Vassiliev invariants, electronic publication, http://www.math.toronto.edu/~drorbn/LOP.html#Computations.
[Kontsevich] ^ M. Kontsevich, unpublished.
[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.
[Zagier_01] ^ D. Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology 40(5) (2001) 945-960.