Knot Floer Homology and Categorification

With the goal of better understanding the connections between knot homology theories arising from categorification and from Heegaard Floer homology, we present a self-contained construction of knot Floer homology in the language of HOMFLY-PT homology. Using the cube of resolutions for knot Floer homology defined by Ozsváth and Szabó;, we first give a purely algebraic proof of invariance that does not depend on Heegaard diagrams, holomorphic disks, or grid diagrams. Then, taking Khovanov's HOMFLY-PT homology as our model, we define a category of twisted Soergel bimodules and construct a braid group action on the homotopy category of complexes of twisted Soergel bimodules. We prove that the category of twisted Soergel bimodules categorifies the Hecke algebra with an extra indeterminate and its inverse adjoined. The braid group action, which is defined via twisted Rouquier complexes, is simultaneously a natural extension of the knot Floer cube of resolutions and a mild modification of the action by Rouquier complexes used by Khovanov in defining HOMFLY-PT homology. Finally, we introduce an operation Qu to play the role that Hochschild homology plays in HOMFLY-PT homology. We conjecture that applying Qu to the twisted Rouquier complex associated to a braid produces the knot Floer cube of resolutions chain complex associated to its braid closure. We prove a partial result in this direction