Linear Algebra for Khovanov Homology

Question

I am studying this lecture notes on Khovanov Homology by Turner. While I get the big picture , there are some things in the Linear Algebra part (having mostly to do with grading) which I cannot understand. So here are some questions which I would like an answer or some reference:

1. Page $7$:When one writes $W=\oplus_mW^m $, does the $m$ stand for the degree ?If that is so, would it be correct to write $V=V^{-1}\oplus V^{1}$ where $V^{-1}=<x>$ and $V^1=<1> ?$
2. Page $8$: How do negative degrees work? Since $\deg (x)=-1$ and $\deg(1)=1$ it would be sensible that $\deg (1\otimes x)=0$ and $\deg (x\otimes x)=-2$ (which is somewhat implied by excerise $3.1$)
3. Page $8$: I cannot make sense of the shift operator and would like a concrete example.If I understand correctly if we have $V=V^{-1}\oplus V^1=<x> \oplus <1>$ and qdim$V=q^{-1}+q$ while $V\{1\}=V^{-2}\oplus V^{0}$ so qdim${V\{1\}}=q^{-2}+1=q^{-1}(q^{-1}+q)=q^{-1}$qdim$V$. But this contradicts the text , where qdim$V\{l\}=q^l$qdim$V$.

Appart from those questions, a good reference for those kinds of stuff would be really appreciated.

Answer 1

1. You are correct. The notation $W^m$ means the summand of $W$ that is in degree $m$ (where degree means polynomial/Jones/quantum degree or grading). It is correct to write $V$ as $V=V^{-1}\oplus V^1$ where $V^{-1}=\langle x \rangle$ and $V^1=\langle 1 \rangle$.
2. Again you are correct. When considering a tensor product of elements, we add the degrees.
3. Turner defines $W\{\ell\}^m = W^{m-\ell}$. So $V\{1\}^2 = V^{2-1} = V^1 = \langle 1 \rangle$ and $V\{1\}^0 = V^{0-1}=V^{-1} = \langle x \rangle$. Thus $\operatorname{qdim} V\{1\} = V^0 \oplus V^2$, and $\operatorname{qdim} V\{1\} = 1+q^2 = q(q^{-1}+q)=q\; \operatorname{qdim} V.$

Other good introductory papers on Khovanov homology include

• Bar-Natan. On Khovanov's categorification of the Jones polynomial.
• Viro. Remarks on the definition of Khovanov homology.
• Turner. A hitchhiker's guide to Khovanov homology.
• Kauffman. Khovanov homology.