In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification...

Mikhail Khovanov constructed a certain chain complex for knots and links and showed that the homology induced from it is a knot invariant (see Khovanov homology)...

He is known for introducing Khovanov homology for links, which was one of the first examples of categorification. Khovanov graduated from Moscow State...

Flat cohomology Floer homology Galois cohomology Group cohomology Hochschild cohomology Intersection cohomology Khovanov homology Lie algebra cohomology...

Borel–Moore homology Cellular homology Cyclic homology Hochschild homology Floer homology Intersection homology K-homology Khovanov homology Morse homology Persistent...

distinguish the unknot from all other knots, such as Khovanov homology and knot Floer homology. Other invariants can be defined by considering some integer-valued...

invariant. (Their homologies satisfy similar formal properties to the combinatorially-defined Khovanov homology.) These homologies are closely related...

research interests include knot theory, finite type invariants, and Khovanov homology. Bar-Natan earned his B.Sc. in mathematics at Tel Aviv University...

known to be in both NP and co-NP. It is known that knot Floer homology and Khovanov homology detect the unknot, but these are not known to be efficiently...

understand a concept in mathematics called Khovanov homology. Developed by Mikhail Khovanov around 2000, Khovanov homology provides a tool in knot theory, the...

Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket...

Jones polynomial), and have been used in describing the properties of Khovanov homology with respect to tangle composition. Any subfactor planar algebra provides...

Mrowka. Jacob Rasmussen later gave a purely combinatorial proof using Khovanov homology, by means of the s-invariant. Kronheimer, P. B.; Mrowka, T. S. (1993)...

Sarkar 2009). Khovanov homology detects the unknot according to a result of Kronheimer and Mrowka. The complexity of Khovanov homology at least as high...

{\displaystyle S} is constructed from f disjoint disks by attaching d bands. The homology group H 1 ( S ) {\displaystyle H_{1}(S)} is free abelian on 2g generators...

polynomial has been shown to be related to Floer homology. The graded Euler characteristic of the knot Floer homology of Peter Ozsváth and Zoltan Szabó is the...

the first homology (with integer coefficients) of X, denoted H 1 ( X ) {\displaystyle H_{1}(X)} . The transformation t acts on the homology and so we...

Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket...

others—studied knots from the point of view of the knot group and invariants from homology theory such as the Alexander polynomial. This would be the main approach...

Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket...

Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket...

used in extending to tangles some properties of Jones polynomial and Khovanov homology of alternating links. An alternating planar algebra is an oriented...

characteristic of the Khovanov homology is equal to the original Jones polynomial. The generators for the chain complex of the Khovanov homology are states of...

Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket...

understand a concept in mathematics called Khovanov homology. Developed by Mikhail Khovanov around 2000, Khovanov homology provides a tool in knot theory, the...

extended to tangles some celebrated results of knot theory about the Khovanov homology and the Jones polynomial. During his career as a mathematician, his...

doi:10.1016/j.ic.2013.03.007, S2CID 17127693 Bar-Natan, Dror (2005), "Khovanov's homology for tangles and cobordisms", Geom. Topol., 9 (3): 1443–1499, arXiv:math/0410495...

Cobordism of Knots. Osaka J. Math. 3, p. 257–267, 1966. Jacob Rasmussen: Khovanov homology and the slice genus. Inv. Math. 182, p. 419–447, 2010. For the orientation...

Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket...

group Z; this follows because the abelianization agrees with the first homology group, which can be easily computed. The knot group (or fundamental group...