---
tag: 64GEW8D5G0
kind: definition
name: Khovanov homology
msc: [57K18]
defines: XB24W24B20
uses: [1A5WRCNTM3, JYKVYS4N32]
sources:
  - url: https://en.wikipedia.org/wiki/Khovanov_homology
---

## Definition

Let $D$ be a diagram of an oriented tame [[link|1A5WRCNTM3]] $L$, 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|JYKVYS4N32]]) 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 $j$.

## Remarks

The map $d$ satisfies $d^2 = 0$ and preserves the quantum grading.
Up to isomorphism, $Kh^{i,j}(L)$ does not depend on the diagram $D$.
Example: the round diagram of the [[unknot|1TQ06R4Q66]] has $n = 0$, so the complex is $A$ in degree $0$ and $Kh$ is $\Z$ in bidegrees $(0, 1)$ and $(0, -1)$.
Its graded Euler characteristic $q + q^{-1}$ is the unnormalized Jones polynomial of the unknot.
In general $\sum_{i,j} (-1)^i q^j \rank Kh^{i,j}(L) = (q + q^{-1}) V_L(t)$ at $t^{1/2} = -q$, where $V_L$ is the [[Jones polynomial|4SPC17FPXE]].
Non-example: the homology of the complex $\bigoplus_v C_v$ with the signs omitted is not defined, because without signs the faces of the cube commute and $d^2 \ne 0$.