Sutured Khovanov homology, Hochschild homology, and the Ozsváth-Szabó spectral sequence

Abstract

Abstract. In [16], Khovanov-Seidel constructed a faithful action of the (m + 1)-strand braid group, B m+1 , on the derived category of left modules over a quiver algebra, Am. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.

A rank inequality for the annular Khovanov homology of 2–periodic links

“…ForL and L as above, rk F 2 AKh j,k (L) ≤ rk F 2 AKh 2j−k,k (L). This result generalizes [8], where Cornish proves a rank inequality between the nextto-top sl 2 weight space gradings in the annular Khovanov homologies of a 2-periodic link and that of its quotient by employing the Lipshitz-Treumann spectral sequence [13] with Auroux, Grigsby, and Wehrli's identification of the next-to-top winding grading with the Hochschild homology of a Khovanov-Seidel bimodule [3]. He remarks that by using Beliakova-Putyra-Wehrli's generalization of Auroux, Grigsby, and Wehrli's work [5], one should be able to prove similar rank inequalities in other sl 2 weight space gradings, but that this requires checking the Lipshitz-Treumann algebraic conditions for larger dg algebras.…”

Annular Khovanov homology and knotted Schur–Weyl representations

“…Then the exterior current algebra sl 2 (∧) acts linearly on SKh(L), and the isomorphism class of this representation is an annular link invariant. 2 Asaeda-Przytycki-Sikora [1] in fact introduced a version of Khovanov homology for links in thickened oriented surfaces F × I. The annular case F = A was explored further by L. Roberts in [33], who related it to Heegaard Floer knot homology as in [27] (see Sec.…”

“…3 For fixed n ∈ Z + , Chen-Khovanov and Brundan-Stroppel introduce a further grading C(n) = ⊕ n k=0 C(n, k) on the category C. A more precise version of the conjecture relates the Hochschild homology of the bimodule associated to the category C(n, k) with a graded summand SKh(L; −n + 2k) ⊆ SKh(L). In [2], the conjecture is proved in the k = 1 case. left and right adjoint to one another, and the adjunction 2-morphisms give rise to endomorphisms e, f :…”

Quantum link homology via trace functor I

“…When q = 1, Theorem C proves Conjecture 1.1 from [AGW15] and motivates us to call our new link invariant the quantum annular link homology.…”

“…The 0th quantum Hochschild homology of A n can be computed by hands, which is all one needs to understand the construction of our invariant. However, computation of higher Hochschild homology is needed to identify the invariant with the total Hochschild homology of Chen-Khovanov bimodules, as conjectured in [AGW15]. For that we reprove the Keller's result for quantum and-more generally-for twisted Hochschild homology of an algebra A by identifying the latter with quantum Hochschild-Mitchell homology of the category of finite dimensional representations of A, twisted by an appropriate endofunctor.…”