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 [45], 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 H(b, q) ⊕ [special characters omitted][ℓ, ℓ⁻¹], where H( b, q) is the Hecke algebra. 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 [21]. 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.