Fixing the functoriality of Khovanov homology

By David A. Clark, S. Morrison, Kevin Walker. 2007

126 citations recorded by the source corpus.

We describe a modification of Khovanov homology [13], in the spirit of Bar-Natan [2], which makes the theory properly functorial with respect to link cobordisms. This requires introducing “disorientations” in the category of smoothings and abstract cobordisms between them used in Bar-Natan’s definition. Disorientations have “seams” separating oppositely oriented regions, coming with a preferred normal direction. The seams satisfy certain relations (just as the underlying cobordisms satisfy relations such as the neck cutting relation). We construct explicit chain maps for the various Reidemeister moves, then prove that the compositions of chain maps associated to each side of each of Carter, Rieger and Saito’s movie moves [8; 7] always agree. These calculations are greatly simplified by following arguments due to Bar-Natan and Khovanov, which ensure that the two compositions must agree, up to a sign. We set up this argument in our context by proving a result about duality in Khovanov homology, generalising previous results about mirror images of knots to a “local” result about tangles. Along the way, we reproduce Jacobsson’s sign table [10] for the original “unoriented theory”, with a few disagreements.

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

View Paper: https://www.semanticscholar.org/paper/72e9ddb0fd2c68e4f2aa54c712ae870cfa7aee9f

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Citations in the corpus

Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, methodology): “Clark-Morrison-Walker sl3 foams.”
Functoriality of colored link homologies (2017, methodology): “For N = 2, functorial theories were obtained by Caprau [7], Clark–Morrison–Walker [13] and Blanchet [5] after Khovanov’s original theory turned out to have a sign ambiguity, see Jacobsson [20] and Bar-Natan [2].”
Highest weight categories arising from Khovanov's diagram algebra II: Koszulity (2008, background): “As explained in [SW, §5.2], the construction of the signed versions differs from Khovanov’s construction in that the underlying 2-dimensional TQFT structure is enhanced by working inside the category of disoriented cobordisms in the sense of [CMW].”
Cobordisms of sutured manifolds and the functoriality of link Floer homology (2009, result): “ng it as a categoriﬁcation of the Alexander polynomial. Compare this with the results of Jacobsson [18] and Khovanov [26] that Khovanov homology is a categoriﬁcation of the Jones polynomial, also see [5]. As link Floer homology turns out to be an invariant of based links [25], and in many cases movi”
The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality (2014, methodology): “In the sl2 case, this construction is originally due to Blanchet [3], following earlier work of Clark-Morrison-Walker [16] on the functoriality of Khovanov homology.”
Transverse knots distinguished by knot Floer homology (2007, background): “}) ∪(K′ ×{1}) (where −K is K with the orientation reversed), there is a map Φ(Σ) : HFK−(K) →HFK−(K′) which depends only on the isotopy class of Σ. (The analogous theorem is true for Khovanov homology [15, 20].) If Conjecture 11 were true, the transverse invariant θ(T) of a transverse knot T of topol”