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The $CFK^\infty$ Type of Almost L-space Knots

Fraser Binns

TL;DR

This work classifies the $\mathrm{CFK}^{\infty}$ type of almost $L$-space knots, proving it must be either a staircase$\oplus$box complex or an almost staircase of type 1 or type 2, with further refinements linking type 1 to certain cabling constructions. Leveraging immersed-curve techniques and bordered Floer theory, it translates rank conditions on large surgeries into precise restrictions on $\mathrm{CFK}^{\infty}$, and derives broad topological consequences such as fiberedness, strong quasi-positivity, and detection results for cables and certain links. The results extend the classic $L$-space knot classification to the almost $L$-space setting, providing a framework to study related properties and conjectures in Heegaard Floer theory.

Abstract

Heegaard Floer homology and knot Floer homology are powerful invariants of 3-manifolds and links respectively. L-space knots are knots which admit Dehn surgeries to 3-manifolds with Heegaard Floer homology of minimal rank. In this paper we study almost L-space knots, which are knots admitting large Dehn surgeries to 3-manifolds with Heegaard Floer homology of next-to-minimal rank. Our main result is a classification of the $CFK^\infty$ type of almost L-space knots. As corollaries we show that almost L-space knots satisfy various topological properties, including some given by Baldwin-Sivek. We also give some new cable link detection results.

The $CFK^\infty$ Type of Almost L-space Knots

TL;DR

This work classifies the type of almost -space knots, proving it must be either a staircasebox complex or an almost staircase of type 1 or type 2, with further refinements linking type 1 to certain cabling constructions. Leveraging immersed-curve techniques and bordered Floer theory, it translates rank conditions on large surgeries into precise restrictions on , and derives broad topological consequences such as fiberedness, strong quasi-positivity, and detection results for cables and certain links. The results extend the classic -space knot classification to the almost -space setting, providing a framework to study related properties and conjectures in Heegaard Floer theory.

Abstract

Heegaard Floer homology and knot Floer homology are powerful invariants of 3-manifolds and links respectively. L-space knots are knots which admit Dehn surgeries to 3-manifolds with Heegaard Floer homology of minimal rank. In this paper we study almost L-space knots, which are knots admitting large Dehn surgeries to 3-manifolds with Heegaard Floer homology of next-to-minimal rank. Our main result is a classification of the type of almost L-space knots. As corollaries we show that almost L-space knots satisfy various topological properties, including some given by Baldwin-Sivek. We also give some new cable link detection results.
Paper Structure (10 sections, 32 theorems, 7 equations, 6 figures)

This paper contains 10 sections, 32 theorems, 7 equations, 6 figures.

Table of Contents

  1. Introduction
  2. A Review of Heegaard Floer Homology
  3. Heegaard Floer homology for closed $3$-manifolds
  4. Knot Floer homology
  5. Chain Complexes for almost $L$-space knots
  6. Some Remarks on Almost $L$-spaces and almost $L$-space knots
  7. $\mathop{\mathrm{CFK}}\nolimits^\infty$ of almost $L$-space knots as a bigraded vector space without Maslov gradings
  8. Maslov Gradings
  9. The filtered chain homotopy type of $\mathop{\mathrm{CFK}}\nolimits^\infty$
  10. Applications

Key Result

Theorem 1.3

Let $K$ be an almost $L$-space knot. Then $\mathop{\mathrm{CFK}}\nolimits^\infty(K)$ has the filtered chain homotopy type of one of the following complexes;

Figures (6)

  • Figure 1: A staircase complex, as in part \ref{['staircasebox']} of Theorem \ref{['infinityclassification']}. The horizontal direction indicates the $U$ grading and the vertical direction indicates the $A$ grading.
  • Figure 2: A box complex, as in part \ref{['staircasebox']} of Theorem \ref{['infinityclassification']}. The horizontal direction indicates the $U$ grading and the vertical direction indicates the $A$ grading. Note the arrows are of length one.
  • Figure 3: An almost staircase complex of type $1$, as in part \ref{['almoststaircase']} of Theorem \ref{['infinityclassification']}. . The horizontal direction indicates the $U$ grading and the vertical direction indicates the $A$ grading.
  • Figure 4: An almost staircase complex of type $2$, as in part \ref{['almoststaircase']} of Theorem \ref{['infinityclassification']}. . The horizontal direction indicates the $U$ grading and the vertical direction indicates the $A$ grading.
  • Figure 5: A pair of exact triangles we will use repeatedly in this section and the next.
  • ...and 1 more figures

Theorems & Definitions (66)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4: baldwin2022characterizing
  • Corollary 1.5
  • Corollary 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 56 more