\( \def\bbN{{\mathbb N}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calT{{\mathcal T}} \) © | Dror Bar-Natan: Classes: 2022-23: Fast Computations in Knot Theory:

Homework Assignment 1

Solve and submit the following problems.

Problem 1.

  1. Prove that the Jones polynomial satisfies the "skein relation" in the figure below.
  2. Show that this skein relation, along with the value of $J$ on the unknot, determine $J$. Phrase your answer as "here is an algorithm to compute $J$ on any knot/link, using only the skein relation and the value on the unknot".

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 $S^2$. (This one is tough. Don't feel too bad if you don't succeed).

Problem 4. Use the programs we wrote in class to compute the Jones polynomial of the Conway Knot:

An excellent solution will include a printout of the Mathematica notebook in which you carried out the computation.

Due date. This assignment is due on Monday July 3 at 11:59pm. Submission is on Google Classroom.