Khovanov homology

Definition

Let $D$ be a diagram of an oriented tame link $L$, with crossings ordered $1, \ldots, n$, of which $n_+$ are positive and $n_-$ are negative. Let $A = \Z[x]/(x^2)$, graded by $\deg 1 = 1$ and $\deg x = -1$. For $v \in \set{0,1}^n$, let $D_v$ be the diagram where crossing $i$ gets the $0$-smoothing (A-smoothing of the Kauffman bracket) if $v_i = 0$ and the $1$-smoothing otherwise. Let $k_v$ be the number of circles of $D_v$ and $r(v) = v_1 + \cdots + v_n$. Put $C_v = A^{\otimes k_v}$, with quantum grading shifted up by $r(v) + n_+ - 2n_-$. The chain group in homological degree $i$ is $C^i(D) = \bigoplus_{r(v) = i + n_-} C_v$. For an edge $v \to v'$ that changes one $0$ to a $1$, the map $C_v \to C_{v'}$ is:

1. $m \colon A \otimes A \to A$ if two circles merge, with $m(1 \otimes 1) = 1$, $m(1 \otimes x) = m(x \otimes 1) = x$, $m(x \otimes x) = 0$;
2. $\Delta \colon A \to A \otimes A$ if one circle splits, with $\Delta(1) = 1 \otimes x + x \otimes 1$ and $\Delta(x) = x \otimes x$;

and the identity on the other circles. The differential $d \colon C^i(D) \to C^{i+1}(D)$ is the sum of these edge maps, each with a sign $(-1)^{s}$. Here $s$ is the number of $1$s in $v$ before the changed position. The Khovanov homology of $L$ is $Kh^{i,j}(L) = H^i(C(D))$ in quantum degree $j$.