06-1350/Homework Assignment 3

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Solve the following problem and submit your solution in class by November 16, 2006:

Problem. The product of two polynomials is again a polynomial; there must be an analog for that in the world of "polynomial" invariants of knots.

1. Prove that the product of two finite type invariants (of, say, knotted $\Gamma$'s) is again a finite type invariant. Of what type will it be, as a function of the types of the two factors of the product?
2. In what way does the product of finite type invariant induces a map $\Box :{\mathcal {A}}(\Gamma )\to {\mathcal {A}}(\Gamma )\otimes {\mathcal {A}}(\Gamma )$?
3. Describe the map $\Box$ of above in explicit terms. First use the "chords and 4T" description of ${\mathcal {A}}(\Gamma )$, and then the "trivalent diagrams and AS, IHX and STU" description of the same object.
4. Learn somewhere about coalgebras and show that $\Box$ is always coassociative and cocommutative.
5. Learn somewhere about bialgebras (Hopf algebras without an antipode) and show that ${\mathcal {A}}(\bigcirc )$ becomes a commutative associative cocommutative and coassociative bialgebra, if taken with the "connected sum" product and with $\Box$ as a coproduct.

With a tiny bit of further algebra and quoting an old theorem of Milnor and Moore, it follows that ${\mathcal {A}}(\bigcirc )$ is a commutative graded polynomial algebra with finitely many generators at each degree. The dimension of the spaces of generators at degrees up two 12 are known and are denoted $\dim {\mathcal {P}}_{m}$ in the table below, which is reproduced from 06-1350/Class Notes for Thursday October 12:

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