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Knot Floer homology detects genus-one fibred knots. (English) Zbl 1149.57019

Summary: Ozsváth and Szabó 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 and links, obtaining as a corollary that if rational surgery on a knot \(K\) gives the Poincaré homology sphere \(\Sigma(2, 3, 5)\), then \(K\) is the left-handed trefoil knot.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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