06-1350/Class Notes for Thursday October 12

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A Question

Determine the image of the differential operator

$D=d\left(1-H{\frac {\partial }{\partial H}}\right)+H{\frac {\partial }{\partial l}},$

acting on polynomials in the variables $d$, $H$ and $l$. More precisely - determine the quotient of the space of polynomials in $d$, $H$ and $l$ by the image of $D$. 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

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

m	0	1	2	3	4	5	6	7	8	9	10	11	12
$\dim {\mathcal {A}}_{m}^{r}$	1	0	1	1	3	4	9	14	27	44	80	132	232
$\dim {\mathcal {A}}_{m}$	1	1	2	3	6	10	19	33	60	104	184	316	548
$\dim {\mathcal {P}}_{m}$	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 $m$. 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 $m$, $\dim {\mathcal {P}}_{m}>e^{c{\sqrt {m}}}$ (for any fixed $c<\pi {\sqrt {\frac {2}{3}}}$, see [Dasbach_00], [Kontsevich]) and $\dim {\mathcal {A}}_{m}<6^{m}m!{\sqrt {m}}/\pi ^{2m}$ ([Stoimenow_98], [Zagier_01]).

Some References

[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.

[Broadhurst_97] ^  D. J. Broadhurst, Conjectured enumeration of Vassiliev invariants, preprint, September 1997, arXiv:q-alg/9709031.

[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.

[Kneissler_97] ^  J. A. Kneissler, The number of primitive Vassiliev invariants up to degree twelve, preprint, June 1997, arXiv:q-alg/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) 94-114.

[Zagier_01] ^  D. Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology 40(5) (2001) 945-960.