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Register a new userGenerators, relations, and homology for Ozsvath-Szabos Kauffman-states algebras
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We give a generators-and-relations description of differential graded algebras recently introduced by Ozsvath and Szabo for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.
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We define new differential graded algebras A(n,k,S) in the framework of Lipshitz-Ozsvath-Thurstons and Zarevs strands algebras from bordered Floer homology. The algebras A(n,k,S) are meant to be strands models for Ozsvath-Szabos algebras B(n,k,S); indeed, we exhibit a quasi-isomorphism from B(n,k,S) to A(n,k,S). We also show how Ozsvath-Szabos gradings on B(n,k,S) arise naturally from the general framework of group-valued gradings on strands algebras.
We define and study a bigraded knot invariant whose Euler characteristic is the Alexander polynomial, closely connected to knot Floer homology. The invariant is the homology of a chain complex whose generators correspond to Kauffman states for a knot diagram. The definition uses decompositions of knot diagrams: to a collection of points on the line, we associate a differential graded algebra; to a partial knot diagram, we associate modules over the algebra. The knot invariant is obtained from these modules by an appropriate tensor product.
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 construct ternary self-distributive (TSD) objects from compositions of binary Lie algebras, $3$-Lie algebras and, in particular, ternary Nambu-Lie algebras. We show that the structures obtained satisfy an invertibility property resembling that of racks. We prove that these structures give rise to Yang-Baxter operators in the tensor product of the base vector space and, upon defining suitable twisting isomorphisms, we obtain representations of the infinite (framed) braid group. We use these results to construct invariants of (framed) links. We consider examples for low-dimensional Lie algebras, where the ternary bracket is defined by composition of the binary ones, along with simple $3$-Lie algebras, and their applications to some classes of links.
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