Khovanov homology
In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial.
It was developed in the late 1990s by Mikhail Khovanov, then at the University of California, Davis, now at Columbia University.
To any link diagram D representing a link L, we assign the Khovanov bracket [D], a cochain complex of graded vector spaces. This is the analogue of the Kauffman bracket in the construction of the Jones polynomial. Next, we normalise [D] by a series of degree shifts (in the graded vector spaces) and height shifts (in the cochain complex) to obtain a new cochain complex C(D). The cohomology of this cochain complex turns out to be an invariant of L, and its graded Euler characteristic is the Jones polynomial of L.
This definition follows the formalism given in Dror Bar-Natan's 2002 paper.
Let {l} denote the degree shift operation on graded vector spaces—that is, the homogeneous component in dimension m is shifted up to dimension m + l.
Similarly, let [s] denote the height shift operation on cochain complexes—that is, the rth vector space or module in the complex is shifted along to the (r + s)th place, with all the differential maps being shifted accordingly.
Let V be a graded vector space with one generator q of degree 1, and one generator q−1 of degree −1.
Now take an arbitrary diagram D representing a link L. The axioms for the Khovanov bracket are as follows:
[ø] = 0 → Z → 0, where ø denotes the empty link.
[O D] = V ⊗ [D], where O denotes an unlinked trivial component.
[D] = F(0 → [D0] → [D1]{1} → 0)
In the third of these, F denotes the flattening' operation, where a single complex is formed from a double complex by taking direct sums along the diagonals. Also, D0 denotes the 0-smoothing' of a chosen crossing in D, and D1 denotes the 1-smoothing', analogously to the skein relation for the Kauffman bracket. Next, we construct the normalised' complex C(D) = [D][−n−]{n+ − 2n−}, where n− denotes the number of left-handed crossings in the chosen diagram for D, and n+ the number of right-handed crossings.