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Search: All articles in the CJM digital archive with keyword Khovanov homology

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1. CJM Online first

Rushworth, William
Doubled Khovanov homology
We define a homology theory of virtual links built out of the direct sum of the standard Khovanov complex with itself, motivating the name doubled Khovanov homology. We demonstrate that it can be used to show that some virtual links are non-classical, and that it yields a condition on a virtual knot being the connect sum of two unknots. Further, we show that doubled Khovanov homology possesses a perturbation analogous to that defined by Lee in the classical case and define a doubled Rasmussen invariant. This invariant is used to obtain various cobordism obstructions; in particular it is an obstruction to sliceness. Finally, we show that the doubled Rasmussen invariant contains the odd writhe of a virtual knot, and use this to show that knots with non-zero odd writhe are not slice.

Keywords:Khovanov homology, virtual knot concordance, virtual knot theory
Categories:57M25, 57M27, 57N70

2. CJM 2016 (vol 68 pp. 1285)

Ehrig, Michael; Stroppel, Catharina
2-row Springer Fibres and Khovanov Diagram Algebras for Type D
We study in detail two row Springer fibres of even orthogonal type from an algebraic as well as topological point of view. We show that the irreducible components and their pairwise intersections are iterated $\mathbb{P}^1$-bundles. Using results of Kumar and Procesi we compute the cohomology ring with its action of the Weyl group. The main tool is a type $\operatorname D$ diagram calculus labelling the irreducible components in a convenient way which relates to a diagrammatical algebra describing the category of perverse sheaves on isotropic Grassmannians based on work of Braden. The diagram calculus generalizes Khovanov's arc algebra to the type $\operatorname D$ setting and should be seen as setting the framework for generalizing well-known connections of these algebras in type $\operatorname A$ to other types.

Keywords:Springer fibers, Khovanov homology, Weyl group type D

3. CJM 2008 (vol 60 pp. 1240)

Beliakova, Anna; Wehrli, Stephan
Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links
We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples where this invariant is a stronger obstruction to sliceness than the multivariable Levine--Tristram signature.

Keywords:Khovanov homology, colored Jones polynomial, slice genus, movie moves, framed cobordism
Categories:57M25, 57M27, 18G60

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