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 [math]\displaystyle{ w(K) }[/math] denote the writhe (self linking number) of a band knot [math]\displaystyle{ K }[/math].

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

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

[math]\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) }[/math]

and by the initial condition [math]\displaystyle{ H(\bigcirc) }[/math]=1.

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

Question 3.

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