\( \def\bbN{{\mathbb N}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calT{{\mathcal T}} \def\Lim{{\operatorname{Lim}}} \)
© | Dror Bar-Natan: Classes: 2021-22: MAT1350F - Algebraic Knot Theory and Computation: (5) Next: On Beyond Zebra!
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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 $S^2$.

Due date. This assignment is due on Crowdmark by the end of Friday September 24, 2020.