Drorbn at 23:46, 16 October 2006

2006-10-16T23:46:50Z

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{{06-1350/Navigation}}

'''Solve the following problems''' and submit them in class by November 2, 2006:

'''Question 1.''' Let <math>w(K)</math> denote the writhe (self linking number) of a band knot <math>K</math>.
# Is <math>w(K)</math> a finite type invariant? Of what type?
# In what sense is <math>\exp(x\cdot w(K))</math> "made of finite type invariants"?
# Compute the weight system of <math>\exp(x\cdot w(K))</math>.

'''Question 2.''' Recall the HOMFLY-PT polynomial, given by the recursive definition
<center><math>
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></center>
and by the initial condition <math>H(\bigcirc)</math>=1.
# In what sense is <math>H(K)</math> a finite type invariant?
# Compute the weight system of <math>H(K)</math>.

'''Question 3.'''
# Find a concise algorithm to compute the weight system associated with the Lie algebra <math>so(N)</math> in its defining representation.
# Verify that your algorithm indeed satisfies the <math>4T</math> relation.

'''Don't submit''' the following, but do think about it:

'''Question 4.''' Read {{Dror}}'s article [http://www.math.toronto.edu/~drorbn/LOP.html#4CT Lie Algebras and the Four Color Theorem] and convince yourself that it is, after all, a worthless curiosity.

06-1350/Homework Assignment 2 - Revision history

Drorbn at 23:46, 16 October 2006