061350/Class Notes for Thursday October 12

A Question
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 oftenstudied 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
The following table (taken from [BarNatan_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 [BarNatan_96].
m  0  1  2  3  4  5  6  7  8  9  10  11  12 
1  0  1  1  3  4  9  14  27  44  80  132  232  
1  1  2  3  6  10  19  33  60  104  184  316  548  
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 AmirKhosravi. See VasCalc Results  ChordMod4T.
Some Bounds
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:
Theorem. For large , (for any fixed , see [Dasbach_00], [Kontsevich]) and ([Stoimenow_98], [Zagier_01]).
Some References
[BarNatan_95] ^ D. BarNatan, On the Vassiliev knot invariants, Topology 34 (1995) 423472.
[BarNatan_96] ^ D. BarNatan, Some computations related to Vassiliev invariants, electronic publication, http://www.math.toronto.edu/~drorbn/LOP.html#Computations.
[Broadhurst_97] ^ D. J. Broadhurst, Conjectured enumeration of Vassiliev invariants, preprint, September 1997, arXiv:qalg/9709031.
[Dasbach_00] ^ O. T. Dasbach, On the combinatorial structure of primitive Vassiliev invariants III  a lower bound, Comm. in Cont. Math. 24 (2000) 579590, arXiv:math.GT/9806086.
[Kneissler_97] ^ J. A. Kneissler, The number of primitive Vassiliev invariants up to degree twelve, preprint, June 1997, arXiv:qalg/9706022.
[Kontsevich] ^ M. Kontsevich, unpublished.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) 94114.
[Zagier_01] ^ D. Zagier, Vassiliev invariants and a strange identity related to the Dedekind etafunction, Topology 40(5) (2001) 945960.