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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 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
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
The cyclotomic Birman-Murakami-Wenzl (BMW) algebras B_n^k, introduced by R. Haring-Oldenburg, are a generalisation of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (aka Ariki-Koike algebras) and type B knot theory.
In this paper, we generalize the categorifical construction of a quantum group and its canonical basis introduced by Lusztig (cite{Lusztig,Lusztig2}) to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the G