On Khovanov’s categorification of the Jones polynomial

By Dror Bar-Natan. 2002

543 citations recorded by the source corpus.

The working mathematician fears complicated words but loves pictures and diagrams. We thus give a no-fancy-anything picture rich glimpse into Khovanov's novel construction of "the categorification of the Jones polynomial". For the same low cost we also provide some computations, including one that shows that Khovanov's invariant is strictly stronger than the Jones polynomial and including a table of the values of Khovanov's invariant for all prime knots with up to 11 crossings.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dea52335979800372cce5dd53819e2b94f68480f

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

Grid Homology for Knots and Links (2015, background): “For the sake of exposition, we focus on the simplest version, from [103]; see also [8, 222].”
An introduction to diagrammatic algebra and categorified quantum sl(2) (2011, background): “Khovanov homology is a strictly stronger knot invariant [BN02], but even more interestingly, it is a functorial knot invariant [Kho02, Jac04, CMW09, Cap08].”
The Conway knot is not slice (2018, background): “Studying the sliceness of knots which arise as a positive mutant of a slice knot is even trickier: all abelian and all but the subtlest metabelian sliceness obstructions vanish for such a knot, Rasmussen’s s-invariant is conjectured to vanish [BN02], and it is unknown whether any Heegaard Floer slic”
Knot homology via derived categories of coherent sheaves II, $\mathfrak{sl}_{m}$ case (2007, methodology): “... we refer to Tq as the 0 smoothing of T at s and denote it Ts 0, and we refer to T= as the 1 smoothing of T at s and denote it Ts 1. On the other hand, if s is a crossing of type 2, we refer to T= as the 0 smoothing of T at s and denote it Ts 0 and we refer to Tq as the 1 smoothing of T at s and ”
Man and machine thinking about the smooth 4-dimensional Poincaré conjecture (2009, background): “Recently, the Khovanov homology link invariant [24], [5], [6], a categorification of the Jones polynomial, has presented itself as a possibility in the form of Rasmussen’s s-invariant [37].”
Khovanov homology and the slice genus (2004, methodology): “Although we are unable to prove that the s(K) defined here is the same as that determined by Bar-Natan’s conjecture, we do give a fairly general condition (at least for small knots) under which the two agree.”
Annular Khovanov homology and knotted Schur–Weyl representations (2015, methodology): “Now note that φKT = F([KT ]) and φ′KT = F ′([KT ]), where F is the functor defined in § 4 and F ′ denotes Lee’s TQFT [Ras10].”
A concordance invariant from the Floer homology of double branched covers (2005, background): “Alternating knots are H-thin by the work of Lee [16], as are most knots up to ten crossings [2], [13].”
A link invariant from the symplectic geometry of nilpotent slices (2004, background): “q=−t1/2 Kh is known to be a strictly stronger invariant than Vκ [3]; by definition it remains algorithmically computable.”
Knot polynomials and knot homologies (2005, methodology): “The first program for this purpose was written by Bar-Natan [1]; it could be used to compute Kh(L) for links of up to 12 or 13 crossings.”
Knot Diagrammatics (2004, background): “The Khovanov Categorification of the Jones polynomial [6] is important for our concerns.”
Khovanov homology, its definitions and ramifications (2004, methodology): “In the paper [2] by Bar-Natan the term Kauffman bracket is used for another characteristic of a link diagram, which fails to be invariant under the second Reidemeister move.”