06-1350/Homework Assignment 2

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Solve the following problems and submit them in class by November 2, 2006:

Question 1. Let w(K) denote the writhe (self linking number) of a band knot K.

  1. Is w(K) a finite type invariant? Of what type?
  2. In what sense is \exp(x\cdot w(K)) "made of finite type invariants"?
  3. Compute the weight system of \exp(x\cdot w(K)).

Question 2. Recall the HOMFLY-PT polynomial, given by the recursive definition

Failed to parse (unknown function\overcrossing): q^{N/2}H\left(\overcrossing\right)-q^{-N/2}H\left(\undercrossing\right)=(q^{1/2}-q^{-1/2})H\left(\smoothing\right)

and by the initial condition H(\bigcirc)=1.

  1. In what sense is H(K) a finite type invariant?
  2. Compute the weight system of H(K).

Question 3.

  1. Find a concise algorithm to compute the weight system associated with the Lie algebra so(N) in its defining representation.
  2. Verify that your algorithm indeed satisfies the 4T relation.

Don't submit the following, but do think about it:

Question 4. Read Dror's article Lie Algebras and the Four Color Theorem and convince yourself that it is, after all, a worthless curiosity.