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| In [[mathematics]], '''Khovanov homology''' is an invariant of oriented [[knot (mathematics)|knots and links]] that arises as the [[homology (mathematics)|homology]] of a [[chain complex]]. It may be regarded as a [[categorification]] of the [[Jones polynomial]].
| | Myrtle Benny is how I'm called and I feel comfy when people use the full name. Years ago we moved to North Dakota. To play baseball is the hobby he will by no means quit performing. Since she was 18 she's been working as a receptionist but her promotion by no means comes.<br><br>My web site [http://www.ubi-cation.com/ubication/node/6056 http://www.ubi-cation.com/ubication/node/6056] |
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| It was developed in the late 1990s by [[Mikhail Khovanov]], then at the [[University of California, Davis]], now at [[Columbia University]].
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| ==Overview==
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| To any link diagram ''D'' representing a [[knot (mathematics)|link]] ''L'', we assign the '''Khovanov bracket''' '''<nowiki>[</nowiki>'''''D'''''<nowiki>]</nowiki>''', a [[chain complex]] of [[graded vector space]]s. This is the analogue of the [[Kauffman bracket]] in the construction of the [[Jones polynomial]]. Next, we normalise '''<nowiki>[</nowiki>'''''D'''''<nowiki>]</nowiki>''' by a series of degree shifts (in the [[graded vector space]]s) and height shifts (in the [[chain complex]]) to obtain a new chain complex '''C'''(''D''). The [[homology (mathematics)|homology]] of this chain complex turns out to be an [[invariant (mathematics)|invariant]] of ''L'', and its graded [[Euler characteristic]] is the Jones polynomial of ''L''.
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| ==Definition==
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| (This definition follows the formalism given in [[Dror Bar-Natan]]'s paper.)
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| 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''.
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| Similarly, let <nowiki>[</nowiki>''s''<nowiki>]</nowiki> denote the ''height shift'' operation on chain complexes—that is, the ''r''th [[vector space]] or [[Module (mathematics)|module]] in the complex is shifted along to the (''r'' + ''s'')th place, with all the [[differential (mathematics)|differential maps]] being shifted accordingly.
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| Let ''V'' be a graded vector space with one generator ''q'' of degree 1, and one generator ''q''<sup>−1</sup> of degree −1.
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| Now take an arbitrary diagram ''D'' representing a link ''L''. The axioms for the '''Khovanov bracket''' are as follows:
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| # '''<nowiki>[</nowiki>'''''ø'''''<nowiki>]</nowiki>''' = 0 → '''Z''' → 0, where ø denotes the empty link.
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| # '''<nowiki>[</nowiki>'''O ''D'''''<nowiki>]</nowiki>''' = ''V'' ⊗ '''<nowiki>[</nowiki>'''''D'''''<nowiki>]</nowiki>''', where O denotes an unlinked trivial component.
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| # '''<nowiki>[</nowiki>'''''D'''''<nowiki>]</nowiki>''' = '''F'''(0 → '''<nowiki>[</nowiki>'''''D<sub>0</sub>'''''<nowiki>]</nowiki>''' → '''<nowiki>[</nowiki>'''''D<sub>1</sub>'''''<nowiki>]</nowiki>'''{1} → 0)
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| 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, ''D''<sub>0</sub> denotes the `0-smoothing' of a chosen crossing in ''D'', and ''D''<sub>1</sub> denotes the `1-smoothing', analogously to the [[skein relation]] for the Kauffman bracket.
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| Next, we construct the `normalised' complex '''C'''(''D'') = '''<nowiki>[</nowiki>'''''D'''''<nowiki>]</nowiki>'''<nowiki>[</nowiki>−''n''<sub>−</sub><nowiki>]</nowiki>{''n''<sub>+</sub> − 2''n''<sub>−</sub>}, where ''n''<sub>−</sub> denotes the number of left-handed crossings in the chosen diagram for ''D'', and ''n''<sub>+</sub> the number of right-handed crossings.
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| The '''Khovanov homology''' of ''L'' is then defined as the homology '''H'''(''L'') of this complex '''C'''(''D''). 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'''(''L'') 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.
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| In 2006 [[Dror Bar-Natan]] developed a computer program to calculate the Khovanov homology (or category) for any knot.<ref>New Scientist 18 Oct 2008</ref>
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| ==Related theories==
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| 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 reprove a result only demonstrated using [[gauge theory]] and its cousins: [[Jacob Rasmussen]]'s new proof of a theorem of Kronheimer and Mrowka, formerly known as the [[Milnor conjecture (topology)|Milnor conjecture]] (see below). Conjecturally, there is a [[spectral sequence]] relating Khovanov homology with the [[Floer homology|knot Floer homology]] of [[Peter Ozsváth]] and [[Zoltán Szabó (mathematician)|Zoltán Szabó]] (Dunfield et al. 2005). Another spectral sequence (Ozsváth-Szabó 2005) relates a variant of Khovanov homology with the Heegaard Floer homology of the branched [[Double cover (topology)|double cover]] along a knot. A third (Bloom 2009) converges to a variant of the monopole Floer homology of the branched double cover.
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| Khovanov homology is related to the representation theory of the [[Lie algebra]] sl<sub>2</sub>. Mikhail Khovanov and Lev Rozansky have since defined [[cohomology]] theories associated to sl<sub>''n''</sub> for all ''n''. In 2003, [[Catharina Stroppel]] extended Khovanov homology to an invariant of tangles (a categorified version of Reshetikhin-Turaev invariants) which also generalizes to sl<sub>''n''</sub> for all ''n''.
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| 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]].
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| ==The relation to link (knot) polynomials==
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| At [[International Congress of Mathematicians]] in 2006 Mikhail Khovanov provided the following explanation for the relation to knot polynomials from the view point of Khovanov homology.
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| The [[skein relation]] for three links <math>L_1,L_2</math> and <math>L_3</math> is described as
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| :<math>\lambda P(L_1)-\lambda^{-1}P(L_2)=(q-q^{-1})P(L_3).</math>
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| Substituting <math>\lambda=q^n,\ n\le0</math> leads to a link polynomial invariant <math>P_n(L)\in\mathbb{Z}[q,q^{-1}]</math>, normalized so that for <math>n > 0</math>
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| :<math>P_n(unknot)=q^{n-1}+q^{n-3}+\cdots+q^{1-n}</math>
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| and <math>P_0(unknot)=1</math>. For <math>n > 0</math> the polynomial <math>P_n(L)</math> can be interpreted via the [[representation theory]] of [[quantum group]] <math>sl(n)</math> and <math>P_0(L)</math> via that of the quantum Lie [[superalgebra]] <math>U_q(gl(1|1))</math>.
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| :The [[Alexander polynomial]] <math>P_0(L)</math> is the [[Euler characteristic]] of a bigraded knot homology theory.
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| :<math>P_1(L)=1</math> is trivial.
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| :The [[Jones polynomial]] is <math>P_2(L)</math> is the Euler characteristic of a bigraded link homology theory.
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| :The entire [[HOMFLY polynomial|HOMFLY-PT polynomial]] is the Euler characteristic of a triply-graded link homology theory.
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| ==Applications==
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| 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.
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| In 2010, [[Peter B. Kronheimer|Kronheimer]] and [[Tomasz Mrowka|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, the [[Jones polynomial#Open problems|Jones polynomial]] may not.
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| ==References==
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| * Mikhail Khovanov, ''A categorification of the Jones polynomial'', [[Duke Mathematical Journal]] 101 (2000) 359–426. {{arxiv|math.QA/9908171}}.
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| * Catharina Stroppel, ''Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors'', [[Duke Mathematical Journal]] 126 (2005) 547–596.
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| * 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|math.QA/0201043}}.
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| * {{ cite arxiv | title = A link surgery spectral sequence in monopole Floer homology | year = 2009 | author = Jonathan M. Bloom | eprint = 0909.0816 }}
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| * {{ cite arxiv | title = The Superpolynomial for Knot Homologies | year = 2005 | author = Nathan M. Dunfield, [[Sergei Gukov]], Jacob Rasmussen | eprint = math.GT/0505662 }}
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| *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 Heegaard Floer homologies for knots.
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| * {{ cite arxiv | title = Link Homology and Categorification | year = 2006 | author = Mikhail Khovanov| eprint = math.GT/0605339 }}
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| {{Reflist}}
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| ==External links==
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| *[http://arxiv.org/pdf/1005.4346.pdf Khovanov homology is an unknot-detector by [[Kronheimer]] and [[Tomasz Mrowka|Mrowka]] ]
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| *[http://www.math.columbia.edu/~khovanov/talks/talkICM2006.pdf Hand-written slides of M. Khovanov's talk]
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| *[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=416503 Khovanov homology on arxiv.org]
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| {{Knot theory}}
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| {{DEFAULTSORT:Khovanov Homology}}
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| [[Category:Homology theory]]
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| [[Category:Knot theory]]
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Myrtle Benny is how I'm called and I feel comfy when people use the full name. Years ago we moved to North Dakota. To play baseball is the hobby he will by no means quit performing. Since she was 18 she's been working as a receptionist but her promotion by no means comes.
My web site http://www.ubi-cation.com/ubication/node/6056