Homework Assignment 3

Signs (like plus and minus, not like in the zodiac) are the least loved part of mathematics. But even signs, when they come in bulks, can make an interesting theory.

Question 1. Recall the definition of "the sign of an edge", $(-1)^\xi:=(-1)^{\sum_{i\lt j}\xi_i}$ when $\xi_j=\star$, that was given in class (following href=http://drorbn.net/mo13/kh.pdf). Prove that it has the property that was claimed in class - namely, that around each face of the $n$-dimensional cube $\{0,1\}^n$ there is an odd number of edges carrying a minus sign (call this "Property O").

Question 2. Can you explain $(-1)^\xi$, as well as the fact that it satisfies property O, by staring at the algebra of differential forms on ${\mathbb R}^n$? (A hint is in the handout at href=http://drorbn.net/mo13/kh.pdf).

Question 3. Prove that if $s_1$ and $s_2$ are sign assignments to the edges of the cube that both satisfy property O, then there is a sign assignment $\sigma$ to the vertices of the cube for which for every edge $\xi$ we have $s_1(\xi)s_2(\xi)=\sigma(\xi_0)\sigma(\xi_1)$, where $\xi_0$ and $\xi_1$ are the vertices at the ends of an edge $\xi$.

Question 4. How is Question 3 relevant to Khovanov homology?

Question 5. Can you rephrase Question 3 as a statement about the homology or the cohomology of a cube or a part of a cube, with coefficients in something?

Question 6. Verify that the $m$ and $\Delta$ of the class of September 30 satify the "funny Frobenius compatibility condition" of the class of September 28.

This assignment is due on Crowdmark by the end of Wednesday October 7, 2020.