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

Homework Assignment 2

Solve and submit the following problems.

Problem 1. Prove that the sign rule explained in class for edges of multi-dimensional cubes, namely $(-1)^\xi = (-1)^{\sum_{i\lt j}\xi_i}$ where $\xi_j=\star$, satisfies the condition stated in class, namely that among the edges around every square there is an odd number of minus signs.

Problem 2.
(a) What is the total dimension of the Khovanov complex of the trefoil knot (in its standard projection as shown in class)? (By "the total dimension" I mean the sum of all the dimensions of all the vector spaces that appear, ignoring gradings/degrees).
(b) What is the total dimension of the Khovanov complex of the knot $4_1$, projected as shown on the right?
(c) Bonus: What is the total dimension of the Khovanov complex of the Conway knot, using the same projections as in the previous assignment?
(d) Extra Bonus: What is the total dimension of the Khovanov complex of the Piccirillo Knot, using the projection (and PD notation) as in the Mathematica notebooks for days 1 and 2?

The following two questions will become easier after Friday's class:

Problem 3. Prove: If $A$ is a complex, $B$ is a subcomplex of $A$, and $H(A/B)=0$, then $H(B)=H(A)$.

Problem 4. Prove that Khovanov homology is invariant under Reidemeister moves of type 1 (note that there are two cases to check, depending on whether the crossing in the kink is positive or negative).

Due date. This assignment is due on Monday July 10 at 11:59pm. Submission is on Google Classroom.