Subsection 7.2 (0PNJ) · ScalingStacks

7.2 Properties of ${\cal H}^{i,j}$ : Euler characteristic, change of orientation

The Kauffman bracket of an oriented link $L$ is equal to the graded Euler characteristic of the cohomology groups ${\cal H}^{i,j}(L),$ as stated in the following proposition.

0PM3

Proposition 27 For an oriented link $L,$

K(L)=\sum_{i,j\in\mathbb{Z}}(-1)^{i}q^{j}{\mathrm{dim}}_{\mathbb{Q}}({\cal H}^{i,j}(L)\otimes\mathbb{Q}),

(154)

where $K(L)$ is the scaled Kauffman bracket (see Section 2.4).

The proof is completely analogous to that of formula (47). $\square$

The statements and proofs of Propositions 22-24 transfer without change to the case of cohomology groups ${\cal H}^{i,j},$ as indicated below.

Let $L$ be an oriented link and $L^{\prime}$ a component of $L.$ Denote by $l$ the linking number of $L^{\prime}$ with its complement $L\setminus L^{\prime}$ in $L.$ Let $L_{0}$ be the link $L$ with the orientation of $L^{\prime}$ reversed.

0PM4

Proposition 28 For $i,j\in\mathbb{Z}$ there is an equality of isomorphism classes of abelian groups

{\cal H}^{i,j}(L_{0})={\cal H}^{i+2l,j+2l}(L).

(155)

0PM5

Proposition 29 Let $K$ and $K_{1}$ be oriented knots and $(-K)$ be $K$ with the reversed orientation. Then

{\cal H}^{i,j}(K\#K_{1})={\cal H}^{i,j}((-K)\#K_{1})

(156)

Similarly to Proposition 24 we can prove

0PM6

Proposition 30 For an oriented link $L$

{\cal H}^{i,j}(L)=0

(157)

if $j+1\equiv{\mathrm{cm}}(L)({\mathrm{mod}}2).$

Mikhail Khovanov

Original source: arXiv:math/9908171v2

7.2 Properties of ℋi,j{\cal H}^{i,j}: Euler characteristic, change of orientation

7.2 Properties of ${\cal H}^{i,j}$ : Euler characteristic, change of orientation