- Khovanov homology
In

mathematics ,**Khovanov homology**is ahomology theory for knots and links. It may be regarded as acategorification of theJones polynomial .It was developed in the late 1990s by

Mikhail Khovanov , then at theUniversity of California, Davis , now atColumbia University .**Overview**To any link "L", we assign the

**Khovanov bracket**"L[ **"**, a] chain complex ofgraded vector space s. This is the analogue of theKauffman bracket in the construction of theJones polynomial . Next, we normalise"L[ **"**by a series of degree shifts (in the] graded vector space s) and height shifts (in thechain complex ) to obtain a new chain complex**C**("L"), whose gradedEuler 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"thvector 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 →"L[ _{0}**"**→] "L[ _{1}**"**{1} → 0)] In the third of these,

**F**denotes the `flattening' operation, where a single complex is formed from adouble complex by taking direct sums along the diagonals. Also, "L_{0}" denotes the `0-smoothing' of a chosen crossing in "L", and "L_{1}" denotes the `1-smoothing', analogously to theskein 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 of3-manifolds and it has been used to prove results previously only demonstrated usinggauge theory , likeJacob Rasmussen 's new proof of the Milnor conjecture (see below). Conjecturally, there is aspectral sequence relating Khovanov homology with the knot Floer homology ofPeter Ozsváth andZoltán 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 sl_{2}. Mikhail Khovanov and Lev Rozansky have since definedcohomology theories associated to sl_{"n"}for all "n". Paul Seidel and Ivan Smith in 2004 exhibited a singly graded piece of the sl_{2}Khovanov homology as a certain Lagrangian intersectionFloer homology ;Ciprian Manolescu has since simplified their construction and shown how to recover the Jones polynomial from his version of theSeidel-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 theslice genus , and is sufficient to prove the Milnor conjecture.**References*** Mikhail Khovanov, "A categorification of the Jones polynomial",

Duke Mathematical Journal 101 (2000) 359–426. 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) 337–370. 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.

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Mikhail Khovanov**— is a professor of mathematics at Columbia University. He earned a PhD[1] in mathematics from Yale University in 1997, where he studied under Igor Frenkel.[2] His interests include knot theory and algebraic topology. He is most well known for the… … Wikipedia**Floer homology**— is a mathematical tool used in the study of symplectic geometry and low dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an… … Wikipedia**Knot invariant**— In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some… … Wikipedia**Conjecture de Milnor (théorie des nœuds)**— Pour la conjecture de Milnor en K théorie algébrique (en), voir Conjecture de Milnor. En théorie des nœuds, la conjecture de Milnor affirme que le 4 genre … Wikipédia en Français**Timeline of category theory and related mathematics**— This is a timeline of category theory and related mathematics. By related mathematics is meant first hand * Homological algebra * Homotopical algebra * Topology using categories, especially algebraic topology * Categorical logic * Foundations of… … Wikipedia**Milnor conjecture (topology)**— For Milnor s conjecture about K theory, see Milnor conjecture. In knot theory, the Milnor conjecture says that the slice genus of the (p,q) torus knot is (p − 1)(q − 1) / 2. It is in a similar vein to the Thom conjecture. It was first proved by… … Wikipedia**Dror Bar-Natan**— (photo by George Bergman) Born January 30, 1966 … Wikipedia**History of knot theory**— For thousands of years, knots have been used for basic purposes such as recording information, fastening and tying objects together. Over time people realized that different knots were better at different tasks, such as climbing or sailing. Knots … Wikipedia**Igor Frenkel**— Igor Borisovich Frenkel (born April 22 1952) is a mathematician working in mathematical physics. In collaboration with James Lepowsky and Arne Meurman he constructed the monster vertex algebra.Born in Leningrad, USSR (Now St. Petersburg, Russia) … Wikipedia**Théorie des nœuds**— Pour les articles homonymes, voir nœud. illustration de la théorie des nœuds La théorie des nœuds est une branche de la topologie qui consiste e … Wikipédia en Français