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