We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard-Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule defined by Khovanov and Seidel.
Auroux, Denis; Grigsby, J. Elisenda; and Wehrli, Stephan M., "On Khovanov-Seidel Quiver Algebras and Bordered Floer Homology" (2011). Mathematics - Faculty Scholarship. 119.
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