061350/Homework Assignment 2
From Drorbn

Solve the following problems and submit them in class by November 2, 2006:
Question 1. Let denote the writhe (self linking number) of a band knot .
 Is a finite type invariant? Of what type?
 In what sense is Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \exp(x\cdot w(K))
"made of finite type invariants"?
 Compute the weight system of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \exp(x\cdot w(K))
.
Question 2. Recall the HOMFLYPT polynomial, given by the recursive definition
and by the initial condition =1.
 In what sense is a finite type invariant?
 Compute the weight system of .
Question 3.
 Find a concise algorithm to compute the weight system associated with the Lie algebra in its defining representation.
 Verify that your algorithm indeed satisfies the 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.