\( \def\bbN{{\mathbb N}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calT{{\mathcal T}} \def\Lim{{\operatorname{Lim}}} \)
© | Dror Bar-Natan: Classes: 2020-21: MAT1350F - Topics in Knot Theory: (5) Next: Homework Assignment 5
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Homework Assignment 4

Question 1. Prove that if two singular knots have the same underlying chord diagrams, then you can get from one to the other by a sequence of 3D motions and crossing flips.

Question 2. As we wrote them in class,

,

the 4T relations have the form $D_1-D_2=D_3-D_4$. They are of course weaker than the 2T relations, which assert that $D_1=D_3$ and $D_2=D_4$. Show that ${\mathcal D}_m/2T$ is spanned by caravans of 1-hump and 2-hump camels:

Hence over a field (only so we can write dim instead of rank), show that $\dim({\mathcal D}_m/2T) = \lfloor m/2\rfloor + 1$.

Question 3. (Do not do!) 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 Friday October 9. Verify that the algorithm you present computes a functional on chord diagrams which satisfies the FI and 4T relations.

This assignment is due on Crowdmark by the end of Wednesday October 14, 2020.