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 $K\subset S^3$ is called a height $m$ knot if $\operatorname{rank} \widehat{HF}(S^3_n(K))-n=m$ for all integers $n\geq 2g(K)-1$, where $\widehat{HF}$ 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 $CFK^\infty(S^3,K)$ 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 $\widehat{HF}$ for the gluing of two height 4 knot complements.