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

TL;DR

This paper classifies the $CFK^ Infty$ type of almost L-space knots, which are knots with large Dehn surgeries resulting in 3-manifolds with near-minimal Heegaard Floer homology, and explores their topological properties.

Contribution

It provides a classification of the $CFK^ Infty$ type for almost L-space knots and derives new topological and cable link detection results.

Findings

Classified the $CFK^ Infty$ type of almost L-space knots.

Showed almost L-space knots satisfy various topological properties.

Provided new results on cable link detection.

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.