On Khovanov-Seidel quiver algebras and bordered Floer homology
  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. 
Denis Auroux, J. Elisenda Grigsby, and Stephan M. Wehrli
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