The Karoubi envelope and Lee’s degeneration of Khovanov homology

By Dror Bar-Natan, S. Morrison. 2006

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We give a simple proof of Lee’s result from [5], that the dimension of the Lee variant of the Khovanov homology of a 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 strong enough to allow for some simplifications that are otherwise unavailable. 57M25; 57M27, 18E05

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

View Paper: https://www.semanticscholar.org/paper/6b3f0b0b4f5aaaccb13c477d4873554567227015

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Citations in the corpus

Deformations of colored sl(N) link homologies via foams (2015, methodology): “To this end, we adapt Bar-Natan and Morrison’s Karoubi envelope technology [2], originally used to give a “local” proof of Lee’s deformation result, to the setting of foams; see Section 2.”
On Khovanov's cobordism theory for SU3 knot homology (2006, background): “We might suggest fixing this ‘problem’ by passing to the Karoubi envelope (see [3] and references therein) of the foam category, which artificially creates images for every projection.”
Functoriality of colored link homologies (2017, methodology): “This approach draws on the combinatorial interpretation of Lee’s deformation provided by Bar-Natan–Morrison [3].”
A categorification of quantum sl(n) (2008, background): “When C is an additive category, the splitting of idempotents allows us to write (b, e) ∈ Kar(C) as im e and we have b ∼= im e⊕ im (1− e), see for example [2].”
The two-color Soergel calculus (2013, background): “Therefore, the associated polynomial of the left-blue-aligned JWm−1 is equal to the product of the first m roots in the s-aligned order, as well as [1] [1] [1] [2] [2] [3] [2] [4] [3] [5] [3] [6] · · · where the final term has denominator [m − 1].”
An oriented model for Khovanov homology (2010, methodology): “We will compute our oriented version of Lee–Rasmussen homology over Λ = Z[12 ] using the Karoubi completion method of Bar-Natan and Morrison [4].”