[Paper Review] Knot Floer homology detects genus-one fibred knots

Paolo Ghiggini|ArXiv.org|Mar 18, 2006

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper proves that knot Floer homology detects genus-one fibred knots by showing that a genus-one knot is fibred if and only if its top knot Floer homology group is isomorphic to ℤ. The proof combines sutured manifold theory and contact topology, leveraging Gabai’s decomposition and the Ozsváth–Szabó contact invariant. The key result establishes that the left-handed trefoil knot is uniquely determined by rational surgery yielding the Poincaré homology sphere Σ(2,3,5).

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 $Σ(2,3,5)$, then $K$ is the left-handed trefoil knot.

Motivation & Objective

• To prove Ozsváth–Szabó's conjecture that knot Floer homology detects fibred knots in the case of genus-one knots.
• To establish a topological criterion for fibredness using the structure of the top knot Floer homology group.
• To resolve Conjecture $ ilde{I}$, which posits that rational surgery yielding the Poincaré homology sphere implies the knot is the left-handed trefoil.
• To provide a missing step in the classification of knots via Dehn surgery by linking knot Floer homology to surgery invariants.

Proposed method

• Apply sutured manifold decomposition to analyze the knot complement and relate topological properties to knot Floer homology.
• Use Gabai’s theory of sutured manifolds and taut foliations to construct smooth foliations with controlled Euler class and intersection numbers.
• Leverage the Ozsváth–Szabó contact invariant in Heegaard Floer homology to detect non-trivial contact structures on the knot complement.
• Compare Euler classes of foliations associated with positive and negative sutured manifold decompositions to detect asymmetry in the homology class.
• Utilize the surgery exact triangle in Heegaard Floer homology to relate knot Floer homology groups across different surgeries.
• Apply genus-minimising surface arguments in sutured manifolds to compute Euler characteristics and derive genus bounds.

Experimental results

Research questions

• RQ1 Can knot Floer homology detect fibred knots in the genus-one case?
• RQ2 Does the condition $\widehat{HFK}(K,1) = \mathbb{Z}$ imply that a genus-one knot $K$ is fibred?
• RQ3 Does rational surgery yielding the Poincaré homology sphere $\Sigma(2,3,5)$ force the knot to be the left-handed trefoil?
• RQ4 Can the Ozsváth–Szabó contact invariant be used to distinguish non-fibred knots via sutured manifold decomposition?
• RQ5 Is the knot Floer homology of the left-handed trefoil the only possible invariant for knots admitting surgery to $\Sigma(2,3,5)$?

Key findings

• A genus-one knot $K$ is fibred if and only if $\widehat{HFK}(K,1) = \mathbb{Z}$, proving the conjecture for genus-one knots.
• The only fibred knots of genus one are the trefoil knots and the figure-eight knot, so knot Floer homology detects these up to isotopy.
• If rational surgery on a knot $K$ yields the Poincaré homology sphere $\Sigma(2,3,5)$, then $K$ must be the left-handed trefoil knot.
• The knot Floer homology of the left-handed trefoil is the unique invariant for knots that yield $\Sigma(2,3,5)$ via rational surgery.
• The proof establishes that $\kappa_m^+ = \kappa_m^-$ and $e(\mathcal{F}_+, R) \neq e(\mathcal{F}_-, R)$, leading to a contradiction if the knot is non-fibred.
• The contact invariant in Heegaard Floer homology detects the asymmetry in foliation invariants, enabling the distinction of fibred from non-fibred knots.