Khovanovs homology for tangles and cobordisms,

"... ABSTRACT. We propose a framework for unifying the sl(N) Khovanov-Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory which categorifies the HOMFLY polynomial. Moreover, this theory should have an additio ..."

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ABSTRACT. We propose a framework for unifying the sl(N) Khovanov-Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory which categorifies the HOMFLY polynomial. Moreover, this theory should have an additional formal structure of a family of differentials. Roughly speaking, the triply graded theory by itself captures the large N behavior of the sl(N) homology, and differentials capture nonstable behavior for small N, including knot Floer homology. The differentials themselves should come from another variant of sl(N) homology, namely the deformations of it studied by Gornik, building on work of Lee. While we do not give a mathematical definition of the triply graded theory, the rich formal structure we propose is powerful enough to make many non-trivial predictions about the existing knot homologies that can then be checked directly. We include many examples where we can exhibit a likely candidate for the triply graded theory, and these demonstrate the internal consistency of our axioms. We conclude with a detailed study of torus knots, developing a picture which gives new

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Some differentials on Khovanov-Rozansky homology

by Jacob Rasmussen

"... Abstract. We study the relationship between the HOMFLY and sl(N) knot homologies introduced by Khovanov and Rozansky. For each N> 0, we show there is a spectral sequence which starts at the HOMFLY homology and converges to the sl(N) homology. As an application, we determine the KR-homology of kno ..."

Abstract - Cited by 57 (3 self) - Add to MetaCart

Abstract. We study the relationship between the HOMFLY and sl(N) knot homologies introduced by Khovanov and Rozansky. For each N> 0, we show there is a spectral sequence which starts at the HOMFLY homology and converges to the sl(N) homology. As an application, we determine the KR-homology of knots with 9 crossings or fewer. 1.

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Triply-graded link homology and Hochschild homology of Soergel bimodules

by Mikhail Khovanov

"... We trade matrix factorizations and Koszul complexes for Hochschild homology of Soergel bimodules to modify the construction of triplygraded link homology and relate it to Kazhdan-Lusztig theory. Hochschild homology. Let R be a k-algebra, where k is a field, R e = R ⊗k R op be the enveloping algebra ..."

Abstract - Cited by 57 (5 self) - Add to MetaCart

We trade matrix factorizations and Koszul complexes for Hochschild homology of Soergel bimodules to modify the construction of triplygraded link homology and relate it to Kazhdan-Lusztig theory. Hochschild homology. Let R be a k-algebra, where k is a field, R e = R ⊗k R op be the enveloping algebra of R, and M be an R-bimodule (equivalently, a left R e-module). The functor of R-coinvariants associates to M the factorspace MR = M/[R, M], where [R, M] is the subspace of M spanned by vectors of the form rm − mr. We have MR = R ⊗Re M. The R-coinvariants functor is right exact and its i-th derived functor takes M to Tor Re i (R, M). The latter space is also denoted HHi(R, M) and called the i-th Hochschild homology of M. The Hochschild homology of M is the direct sum HH(R, M) def = ⊕ HHi(R, M). i≥0 To compute Hochschild homology, we choose a resolution of the R-bimodule R by projective R-bimodules and tensor the resolution with M:

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Calculating Bar-Natan’s Characteristic Two Khovanov Homology

by Paul Turner , 2004

"... We set up a spectral sequence converging to Bar-Natan’s characteristic two Khovanov homology. The spectral sequence collapses at the E2-page which can be described in terms of groups arising from the action of a certain endomorphism on F2-Khovanov homology. We introduce stable Bar-Natan theory whic ..."

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We set up a spectral sequence converging to Bar-Natan’s characteristic two Khovanov homology. The spectral sequence collapses at the E2-page which can be described in terms of groups arising from the action of a certain endomorphism on F2-Khovanov homology. We introduce stable Bar-Natan theory which we then calculate explicitly. Some consequences are discussed, including some exact sequences in F2-Khovanov homology and a formula for the form of the Khovanov polynomial of a F2homologically thin link.

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THE UNIVERSAL sl3-LINK HOMOLOGY

by Marco Mackaay, Pedro Vaz , 2006

"... ABSTRACT. We define the universal sl3-link homology, which depends on 3 parameters, following Khovanov’s approach with foams. We show that this 3-parameter link homology, when taken with complex coefficients, can be divided into 3 isomorphism classes. The first class is the one to which Khovanov’s o ..."

Abstract - Cited by 27 (6 self) - Add to MetaCart

ABSTRACT. We define the universal sl3-link homology, which depends on 3 parameters, following Khovanov’s approach with foams. We show that this 3-parameter link homology, when taken with complex coefficients, can be divided into 3 isomorphism classes. The first class is the one to which Khovanov’s original sl3-link homology belongs, the second is the one studied by Gornik in the context of matrix factorizations and the last one is new. Following an approach similar to Gornik’s we show that this new link homology can be described in terms of Khovanov’s original sl2-link homology. 1.

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Khovanov homology for virtual knots with arbitrary coefficients

by Vassily Olegovich Manturov - AC. SCI. IZVESTIYA (MATHEMATICS , 2008

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THE KAROUBI ENVELOPE AND LEE’S DEGENERATION OF KHOVANOV HOMOLOGY

by Dror Bar-natan, Scott Morrison

"... Abstract. We give a simple proof of Lee’s result from [5], that the dimension of the Lee variant of the Khovanov homology of an c-component link is 2 c, regardless of the number of crossings. Our method of proof is entirely local and hence we can state a Lee-type theorem for tangles as well as for k ..."

Abstract - Cited by 21 (1 self) - Add to MetaCart

Abstract. We give a simple proof of Lee’s result from [5], that the dimension of the Lee variant of the Khovanov homology of an c-component link is 2 c, regardless of the number of crossings. Our method of proof is entirely local and hence we can state a Lee-type theorem for tangles as well as for knots and links. Our main tool is the “Karoubi envelope of the cobordism category”, a certain enlargement of the cobordism category which is mild enough so that no information is lost yet

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Unoriented topological quantum field theory and link homology

by Vladimir Turaev, Paul Turner

"... We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply.1C1/–dimensional unoriented topological quantum field theories to Bar-Natan’s geometric formalism to define new theories for stable equivalence classes. 57M25, 57R56; 81T40 1 ..."

Abstract - Cited by 20 (1 self) - Add to MetaCart

We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply.1C1/–dimensional unoriented topological quantum field theories to Bar-Natan’s geometric formalism to define new theories for stable equivalence classes. 57M25, 57R56; 81T40 1

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Khovanovs homology for tangles and cobordisms, (2005)

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