06-1350/Class Notes for Tuesday October 10

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Some Questions

Plastic trinions

Question 1. Can you embed a trinion (a.k.a. a sphere with three holes, a pair of pants, or a band theta graph) in ${\mathbb {R} }^{3}$ so that each boundary component would be unknotted yet each pair of boundary components would be knotted? How about, so that at least one pair of boundary components would be knotted?

Dror's Speculation. Yes and yes.

Question 2. A trinion $\gamma$ is embedded in ${\mathbb {R} }^{3}$ so that its boundary is the trivial 3-component link. Does it follow that $\gamma$ is trivial?

Dror's Speculation. No.

Question 3. Suppose two trinions $\gamma _{1}$ and $\gamma _{2}$ are knotted so that the pushforwards $\gamma _{1\star }L$ and $\gamma _{2\star }L$ are equal for any link $L$ which is "drawn" on the parameter space $\Gamma$ of $\gamma _{1}$ and $\gamma _{2}$. Does it follow that $\gamma _{1}$ and $\gamma _{2}$ are equivalent?

Dror's Speculation. I'm clueless.

The standardly embedded strapped trinion

Question 4. A trinion $\gamma$ is embedded in ${\mathbb {R} }^{3}$ so that its "strapped boundary" is equivalent to the strapped boundary of the trivially embedded trinion. Does it follow that $\gamma$ is trivial?

Dror's Speculation. If yes, it will have terrific consequences. If no, it will explain some of the misery we encounter when we deal with "associators". I would really like to understand this one.

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.