# 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 ${\displaystyle w(K)}$ denote the writhe (self linking number) of a band knot ${\displaystyle K}$.

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

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

$\displaystyle q^{N/2}H\left(\overcrossing\right)-q^{-N/2}H\left(\undercrossing\right)=(q^{1/2}-q^{-1/2})H\left(\smoothing\right)$

and by the initial condition ${\displaystyle H(\bigcirc )}$=1.

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

Question 3.

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