$ \def\bbN{{\mathbb N}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calT{{\mathcal T}} \def\Lim{{\operatorname{Lim}}} $

(2)

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Homework Assignment 1

Solve and submit the following problems.

Problem 1. Observe that in $\bbZ/3$, $x+y+z=0$ iff $x,y,z$ are all the same or are all different. Use this to show that the 3-colouring invariant $\lambda(D)$ is always a power of 3 and that it can be computed in polynomial time.

Problem 2. Rather than fixing the Kauffman bracket by using a writhe counter-term, it is tempting to evaluate it at $A=e^{\pi i/3}$, where invariance under R1 holds with no need for a correction. Unfortunately, at $A=e^{\pi i/3}$ the Kauffman bracket of any knot is equal to 1. Prove this.

Problem 3. Prove that the PD notation of a knot diagram determines it as a diagram in $S^2$.

Due date. This assignment is due on Crowdmark by the end of Wednesday September 23, 2020.