06-1350/Homework Assignment 2
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.
- Is w(K) a finite type invariant? Of what type?
- In what sense is "made of finite type invariants"?
- Compute the weight system of .
Question 2. Recall the HOMFLY-PT polynomial, given by the recursive definition
and by the initial condition =1.
- In what sense is H(K) a finite type invariant?
- Compute the weight system of H(K).
- Find a concise algorithm to compute the weight system associated with the Lie algebra so(N) in its defining representation.
- Verify that your algorithm indeed satisfies the 4T relation.
Don't submit the following, but do think about it: