Khovanov's homology for tangles and cobordisms

By Dror Bar-Natan. 2004

569 citations recorded by the source corpus.

We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological. And then a simple application of an appropriate functor (a "TQFT") to our pictures takes them to the familiar realm of complexes of (graded) vector spaces and ordinary homological invariants.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c358fc78802d6f872146793a0343ef40398edca8

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Citations in the corpus

A remark on rasmussen's invariant of knots (2005, background): “When K = Q, h = 0 and t = 1 one gets Lee’s theory [5] and when K = F2, h = 1 and t = 0 one gets Bar-Natan’s theory [1].”
Functoriality of colored link homologies (2017, methodology): “The first equivariant, or universal, link homology can be traced back to Bar-Natan [2] and Khovanov [25] and it encapsulated both Khovanov homology and Lee’s deformation.”
Link homology and Frobenius extensions (2004, methodology): “Although this map from Bar-Natan’s theory to F5 is neither surjective nor injective already for the empty link, his arguments are universal, being instantly adoptable to F5 and any base change of the latter. The next result follows from [ BN ] at once.”
The universal sl3–link homology (2006, methodology): “In the first part of this paper we construct the universalsl3-link homology over Z[a, b, c].”
Homological thickness and stability of torus knots (2005, unknown): “⊕ Cn(D̄ ′ 2)[2]{−2} ր Cn(D̄ ′ 0)[1]{−2} (21)”
On Khovanov's cobordism theory for SU3 knot homology (2006, methodology): “8 The su3 cobordism theory is functorial; in particular the sign problems seen in the su2 case [2, 7, 16] don’t occur.”
An infinite torus braid yields a categorified Jones-Wenzl projector (2010, methodology): “ terms of Temperley-Lieb tangles are rational (rather than polynomial) functions of q. This suggests that the categoriﬁcation Pn of Pn in the universal tangle category TLn constructed by D. Bar-Natan [BN05] should be presented by a semiinﬁnite chain complex. In fact, there are two mutually dual cate”
Fixing the functoriality of Khovanov homology (2007, methodology): “The unoriented versions of these moves are shown in Figure 4 (thanks to Carter and Saito for originally drawing these diagrams!), using the numbering scheme introduced by Bar-Natan in [2].”
The Superpolynomial for Knot Homologies (2005, unknown): “The Superpolynomial for Knot Homologies Nathan M. Dunfield, Sergei Gukov, and Jacob Rasmussen CONTENTS”
Homological algebra of knots and BPS states (2011, background): “Namely, in doubly graded knot homologies constructed from matrix factorizations differentials that relate different theories are in one-to-one correspondence with deformations of the potentials; see Bar-Natan [3], Gornik [19], the first author and Walcher [24], Khovanov [32], Lee [44] and Turner [59”
Annular Khovanov homology and knotted Schur–Weyl representations (2015, methodology): “This follows immediately from the definition and properties of F and from the definitions of [P(L)] of (CKh(P(L)), ∂0) given in [5] and [33], respectively.”
The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality (2014, background): “The invariant JτK should be viewed as the ‘universal’ (undeformed) sln link invariant, akin to the construction given in [1] for sl2 link homology.”
Categorification of the Jones-Wenzl Projectors (2010, methodology): “The results in [15] fit in the context of categorification of the Temperley– Lieb algebra; see [16] and [4] (also [6] and [24]).”
A slice genus lower bound from sl(n) Khovanov–Rozansky homology (2007, methodology): “all also be postponing some of the 1-handle moves, and some of Reidemeister I and II moves as well. Suppose that Sis a composition of elementary cobordisms representing a connected surface Σ Σ → R3 ×[0,1] ∂Σ = K∪U,K→ R3 ×{0},U→ R3 ×{1} where Sbegins with a diagram Dof the knot Kand ends with the 0-c”
Deformations of colored sl(N) link homologies via foams (2015, background): “5 Karoubi envelope technology In his famous paper [1], Bar-Natan shows that Khovanov homology can be constructed locally, by working in the homotopy category of chain complexes over a certain .”