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 [math]\displaystyle{ \Gamma }[/math]'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 [math]\displaystyle{ \Box:{\mathcal A}(\Gamma)\to{\mathcal A}(\Gamma)\otimes{\mathcal A}(\Gamma) }[/math]?
3. Describe the map [math]\displaystyle{ \Box }[/math] of above in explicit terms. First use the "chords and 4T" description of [math]\displaystyle{ {\mathcal A}(\Gamma) }[/math], and then the "trivalent diagrams and AS, IHX and STU" description of the same object.
4. Learn somewhere about coalgebras and show that [math]\displaystyle{ \Box }[/math] is always coassociative and cocommutative.
5. Learn somewhere about bialgebras (Hopf algebras without an antipode) and show that [math]\displaystyle{ {\mathcal A}(\bigcirc) }[/math] becomes a commutative associative cocommutative and coassociative bialgebra, if taken with the "connected sum" product and with [math]\displaystyle{ \Box }[/math] as a coproduct.

With a tiny bit of further algebra and quoting an old theorem of Milnor and Moore, it follows that [math]\displaystyle{ {\mathcal A}(\bigcirc) }[/math] 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 [math]\displaystyle{ \dim{\mathcal P}_m }[/math] 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
[math]\displaystyle{ \dim{\mathcal A}_m^r }[/math]	1	0	1	1	3	4	9	14	27	44	80	132	232
[math]\displaystyle{ \dim{\mathcal A}_m }[/math]	1	1	2	3	6	10	19	33	60	104	184	316	548
[math]\displaystyle{ \dim{\mathcal P}_m }[/math]	0	1	1	1	2	3	5	8	12	18	27	39	55