Knot Floer complexes of generalized L-space knots

In this paper, we study height 4 knots, which generalize L-space knots and almost L-space knots studied by Ozsváth--Szabó, Baldwin--Sivek, and Binns. A knot K⊂ S³ is called a height m knot if rank HF(S³ₙ(K))-n=m for all integers n≥ 2g(K)-1, where HF is the hat version of Heegaard Floer homology introduced by Ozsváth--Szabó. In particular, height 0 and height 2 knots are exactly L-space knots and almost L-space knots, respectively. We prove that the knot Floer complex CFK^∞(S³,K) of a height 4 knot K admits a basis that is horizontally and vertically simplified--an hv-basis for short. Moreover, we show that under such an hv-basis, the horizontal and vertical differentials fall into six cases, generalizing the results for L-space knots and almost L-space knots. The hv-basis of the knot Floer complex of K can determine, via an explicit algorithm, the immersed curve of Hanselman-Rasmussen-Watson for the knot complement of K. Consequently, our results facilitate further exploration of computations of HF for the gluing of two height 4 knot complements.