Knot Floer homology detects genus-one fibred knots

By P. Ghiggini. 2006

239 citations recorded by the source corpus.

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

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/a7a549a7219da0f5ce4d07e2fe2a9ec6c50016d3

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Citations in the corpus

The lens space realization problem (2010, methodology): “This corresponds to 5-surgery along a genus one L-space knot, which must be the right-hand trefoil by a theorem of Ghiggini [15].”
Homologically thin, non-quasi-alternating links (2009, background): “Since a trefoil is uniquely determined by its knot Floer homology [5], the result follows.”
Knots, sutures, and excision (2008, background): “G T ] 3 1 Contents 1 Introduction 1 2 Background on monopole Floer homology 3 3 Floer’s excision theorem 19 4 Monopole Floer homology for sutured manifolds 26 5 Knot homology 33 6 Fibered knots 41 7 Instantons 52”
A combinatorial spanning tree model for knot Floer homology (2011, background): “It is known (or soon will be) that monopole knot homology agrees with knot Floer homology, as a result of nearly one thousand pages of work of Taubes [50, 51, 52, 53, 54], Kutluhan–Lee–Taubes [20, 21, 22, 23, 24] and Colin–Ghiggini–Honda [6, 7], combined with work of Lekili [25].”
On the geography and botany of knot Floer homology (2014, methodology): “As with the trefoil, the figure eight—being the other genus one fibered knot in S3—is also characterized by knot Floer homology [22].”
Knot Floer homology detects fibred knots (2007, background): “1 was previously examined in various sporadic cases, and some theoretical evidences were given in [12], but the first real progress was made by Ghiggini in [5], where a strategy to approach this conjecture was proposed, and the special case of genus-one knots in S3 was proved.”
Knot Floer homology detects fibred knots (2006, background): “1 was previously examined in various sporadic cases, and some theoretical evidences were given in [12], but the first real progress was made by Ghiggini in [5], where a strategy to approach this conjecture was proposed, and the special case of genus-one knots in S3 was proved.”
An algorithm for computing some Heegaard Floer homologies (2006, background): “Moreover, knot Floer homology detects the genus ([11]) and fiberedness ([1], [7], [2]) of knots and links in the three-sphere.”
The sutured Floer homology polytope (2008, methodology): “by Ghiggini in [9] for genus one knots, and proceeds along the lines of the OzsváthSzabó proof of ĤFK(K, g(K)) 6= 0, using deep symplectic and contact topology.”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “Ghiggini proved the “if” part for genus one knots [Ghi08], and Ni proved it in general [Ni07].”
Grid Homology for Knots and Links (2015, methodology): “A theorem of Ghiggini [71] (see also Theorem 1.”