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Register a new userSutured Khovanov homology distinguishes braids from other tangles
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We show that the sutured Khovanov homology of a balanced tangle in the product sutured manifold D x I has rank 1 if and only if the tangle is isotopic to a braid.
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We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A x I, has the structure of a bifiltered complex whose filtered chain homotopy type is an invariant of the isotopy class of L in A x I. Using ideas of Ozsvath-Stipsicz-Szabo as reinterpreted by Livingston, we use this structure to define a family of annular Rasmussen invariants that yield information about annular and non-annular cobordisms. Focusing on the special case of annular links obtained as braid closures, we use the behavior of the annular Rasmussen invariants to obtain a necessary condition for braid quasipositivity and a sufficient condition for right-veeringness.
Let L be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of the exterior current algebra of the Lie algebra sl_2. When L is an m-framed n-cable of a knot K in the three-sphere, its sutured annular Khovanov homology carries a commuting action of the symmetric group S_n. One therefore obtains a knotted Schur-Weyl representation that agrees with classical sl_2 Schur-Weyl duality when K is the Seifert-framed unknot.
We define stable homotopy refinements of Khovanovs arc algebras and tangle invariants.
Friedl and Kim show any taut sutured manifold can be realized as a twisted homology product, but their proof gives no practical description of how complicated the realizing representation needs to be. We give a number of results illustrating the relationship between the topology of a taut sutured handlebody and the complexity of a representation realizing it as a homology product.
We review the construction and context of a stable homotopy refinement of Khovanov homology.
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