$ \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}}} $

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Homework Assignment 5

Question 1. Find a simple algorithm to compute the weight system of the Jones polynomial, in the spirit of the algorithm for the weight system of the Conway polynomial presented in class on Wednesday October 14. Verify that the algorithm you present computes a functional on chord diagrams which satisfies the FI and 4T relations.

Question 2. Prove that indeed the co-product $\Box\colon\calD\to\calD\otimes\calD$ defined in class descends modulo the 4T relations to a co-product $\Box\colon\calA\to\calA\otimes\calA$.

Question 3. Verify the "Hopf Axiom": That the map $\Box\colon\calA\to\calA\otimes\calA$ is a morphism of algebras.

This assignment is due on Crowdmark by the end of Wednesday October 21, 2020.