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Register a new userStrands algebras and Ozsvath-Szabos Kauffman-states functor
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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.
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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.
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.
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