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.
Overview To any link diagram D {\displaystyle D} representing a link L {\displaystyle L} , we assign the Khovanov bracket [ D ] {\displaystyle \left[D\right]} , 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 ] {\displaystyle \left[D\right]} 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 ) {\displaystyle C(D)} . The cohomology of this cochain complex turns out to be an invariant of L {\displaystyle L} , and its graded Euler characteristic is the Jones polynomial of L {\displaystyle L} .
Definition This definition follows the formalism given in Dror Bar-Natan's 2002 paper. Let l {\displaystyle l} denote the degree shift operation on graded vector spaces—that is, the homogeneous component in dimension m {\displaystyle m} is shifted up to dimension m + l {\displaystyle m+l} . Similarly, let [ s ] {\displaystyle [s]} denote the height shift operation on cochain complexes—that is, the r {\displaystyle r} th vector space or module in the complex is shifted along to the ( r + s ) {\displaystyle (r+s)} th place, with all the differential maps being shifted accordingly. Let V {\displaystyle V} be a graded vector space with one generator q {\displaystyle q} of degree 1, and one generator q − 1 {\displaystyle q^{-1}} of degree − 1 {\displaystyle -1} . Now take an arbitrary diagram D {\displaystyle D} representing a link L {\displaystyle L} . The axioms for the Khovanov bracket are as follows:
[ ∅ ] = ( 0 → Z → 0 ) {\displaystyle [\emptyset ]=(0\to \mathbb {Z} \to 0)} , where ∅ {\displaystyle \emptyset } denotes the empty link.
[ O D ] = V ⊗ [ D ] {\displaystyle [{\text{O }}D]=V\otimes [D]} , where O denotes an unlinked trivial component.
[ D ] = F ( 0 → [ D 0 ] → [ D 1 ] { 1 } → 0 ) {\displaystyle [D]=\mathbf {F} (0\to [D_{0}]\to [D_{1}]\{1\}\to 0)}
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