Khovanov homology

In mathematics, Khovanov homology is a homology theory for knots and links. 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.

Overview

To any link "L", we assign the Khovanov bracket ["L"] , 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 ["L"] by a series of degree shifts (in the graded vector spaces) and height shifts (in the chain complex) to obtain a new chain complex C("L"), whose graded Euler characteristic is the Jones polynomial of "L".

Definition

(This definition follows the formalism given in Dror Bar-Natan's 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 chain complexes—that is, the "r"th 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 link "L". The axioms for the Khovanov bracket are as follows:
# ["ø"] = 0 → Z → 0, where ø denotes the empty link.
# [O "L"] = "V" ⊗ ["L"] , where O denotes an unlinked trivial component.
# ["L"] = F(0 → ["L₀"] → ["L₁"] {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, "L<sub>0</sub>" denotes the `0-smoothing' of a chosen crossing in "L", and "L₁" denotes the `1-smoothing', analogously to the skein relation for the Kauffman bracket.

Next, we construct the `normalised' complex C("L") = ["L"] [-"n"_-] {"n"₊-2"n"_-}, where "n"_- denotes the number of left-handed crossings in the chosen diagram for "L", and "n"₊ the number of right-handed crossings.

The Khovanov homology of "L" is then defined as the homology H("L") of this complex C("L"), and its graded Euler characteristic turns out to be the Jones polynomial of "L". However, H("L") has been shown to contain more information about "L" than the Jones polynomial, but the exact details are not yet fully understood.

Related theories

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 and it has been used to prove results previously only demonstrated using gauge theory, like Jacob Rasmussen's new proof of the Milnor conjecture (see below). Conjecturally, there is a spectral sequence relating Khovanov homology with the knot Floer homology of Peter Ozsváth and Zoltán Szabó (Dunfield et al. 2005). Another spectral sequence (Ozsváth-Szabó 2005) relates a variant of Khovanov homology with the Heegard Floer homology of the branched double cover along a knot.

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 sl_"n" for all "n". Paul Seidel and Ivan Smith in 2004 exhibited a singly graded piece of the sl₂ Khovanov homology as a certain Lagrangian intersection Floer homology; Ciprian Manolescu has since simplified their construction and shown how to recover the Jones polynomial from his version of the Seidel-Smith invariant.

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.

References

* Mikhail Khovanov, "A categorification of the Jones polynomial", Duke Mathematical Journal 101 (2000) 359&ndash;426. arxiv|archive=math.QA|id=9908171.
* Dror Bar-Natan, [http://dx.doi.org/10.2140/agt.2002.2.337 "On Khovanov's categorification of the Jones polynomial"] , Algebraic and Geometric Topology 2 (2002) 337&ndash;370. arxiv|archive=math.QA|id=0201043.
*
*Ozsváth, Peter and Szabó, Zoltán. On the Heegaard Floer homology of branched double-covers. Adv. Math. 194 (2005), no. 1, 1--33. Also available as [http://arxiv.org/abs/math.GT/0309170 a preprint] . This paper discusses the spectral sequence relating Khovanov and Heegard Floer homologies for knots.