To any link diagramD representing a linkL, we assign the Khovanov bracket, a chain complex of graded vector spaces. This is the analogue of the Kauffman bracket in the construction of the Jones polynomial. Next, we normalise by a series of degree shifts and height shifts to obtain a new chain complex C. The homology of this chain complex turns out to be an invariant of L, and its graded Euler characteristic is the Jones polynomial of L.
Definition
This definition follows the formalism given in Dror Bar-Natan's 2002 paper. Let 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 denote the height shift operation on chain complexes—that is, the rth vector space or module in the complex is shifted along to the 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:
' = 0 → Z → 0, where ø denotes the empty link.
' = V ⊗ ', where O denotes an unlinked trivial component.
' = F
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 = , where n− denotes the number of left-handed crossings in the chosen diagram for D, and n+ the number of right-handed crossings. The Khovanov homology of L is then defined as the homology H of this complex C. It turns out that the Khovanov homology is indeed an invariant of L, and does not depend on the choice of diagram. The graded Euler characteristic of H turns out to be the Jones polynomial of L. However, H has been shown to contain more information about L than the Jones polynomial, but the exact details are not yet fully understood. In 2006 Dror Bar-Natan developed a computer program to calculate the Khovanov homology for any knot.
One of the most interesting aspects of Khovanov's homology is that its exact sequences are formally similar to those arising in the Floer homology of 3-manifolds. Moreover, it has been used to produce another proof of a result first demonstrated using gauge theory and its cousins: Jacob Rasmussen's new proof of a theorem of Peter Kronheimer and Tomasz Mrowka, formerly known as the Milnor conjecture. There is a spectral sequence relating Khovanov homology with the knot Floer homology of Peter Ozsváth and Zoltán Szabó. This spectral sequence settled an earlier conjecture on the relationship between the two theories. Another spectral sequence relates a variant of Khovanov homology with the Heegaard Floer homology of the branched double cover along a knot. A third converges to a variant of the monopole Floer homology of the branched double cover. In 2010 Kronheimer and Mrowka exhibited a spectral sequence abutting to their instanton knot Floer homology group and used this to show that Khovanov Homology detects the unknot. Khovanov homology is related to the representation theory of the Lie algebra sl2. Mikhail Khovanov and Lev Rozansky have since defined cohomology theories associated to sln for all n. In 2003, Catharina Stroppel extended Khovanov homology to an invariant of tangles which also generalizes to sln for all n. Paul Seidel and Ivan Smith have constructed a singly graded knot homology theory using Lagrangian intersection Floer homology, which they conjecture to be isomorphic to a singly graded version of Khovanov homology. Ciprian Manolescu has since simplified their construction and shown how to recover the Jones polynomial from the chain complex underlying his version of the Seidel-Smith invariant.
The Alexander polynomial is the Euler characteristic of a bigraded knot homology theory.
is trivial.
The Jones polynomial is the Euler characteristic of a bigraded link homology theory.
The entire HOMFLY-PT polynomial is the Euler characteristic of a triply graded link homology theory.
Applications
The first application of Khovanov homology was provided by Jacob Rasmussen, who defined the s-invariant using Khovanov homology. This integer valued invariant of a knot gives a bound on the slice genus, and is sufficient to prove the Milnor conjecture. In 2010, Kronheimer and Mrowka proved that the Khovanov homology detects the unknot. The categorified theory has more information than the non-categorified theory. Although the Khovanov homology detects the unknot, it is not yet known if the Jones polynomial does.
