Knot Floer complexes of generalized L-space knots

Abstract

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 HF̂(S³_n(K))−n=m for all integers n≥ 2g(K)−1, where HF̂ 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 HF̂ for the gluing of two height 4 knot complements.

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BibTeX RIS

BibTeX

@misc{zou2026knot,
  title = {{Knot Floer complexes of generalized L-space knots}},
  author = {Shengyu Zou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35305},
  eprint = {2610.35305},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35305},
}

RIS

TY  - UNPB
TI  - Knot Floer complexes of generalized L-space knots
AU  - Shengyu Zou
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35305
UR  - https://arxiv.org/abs/2610.35305
AB  - 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⊂ S^3 is called a height m knot if rank HF̂(S^3_n(K))−n=m for all integers n≥ 2g(K)−1, where HF̂ 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^∞(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 HF̂ for the gluing of two height 4 knot complements.
ER  -

Abstract

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