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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 $H(\bigcirc )$=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.