Knot Floer homology detects genus-one fibred knots

TL;DR

This paper advances the understanding of knot Floer homology by proposing a strategy to detect genus-one fibred knots, and confirms the conjecture for certain cases related to the Poincare homology sphere.

Contribution

It introduces a new approach combining sutured manifold decomposition and contact topology to detect fibred knots, specifically applying it to genus-one knots.

Findings

Proposes a strategy to approach the conjecture that knot Floer homology detects fibred knots.

Shows that rational surgery on a genus-one knot yielding the Poincare homology sphere implies the knot is the left-handed trefoil.

Provides evidence supporting Ozsvath and Szabo's conjecture for genus-one knots.

Abstract

Ozsvath and Szabo conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on Gabai's theory of sutured manifold decomposition and contact topology. We implement this strategy for genus-one knots, obtaining as a corollary that, if rational surgery on a knot $K$ gives the Poincare homology sphere $\Sigma(2,3,5)$, then $K$ is the left-handed trefoil knot.