Knot Floer homology detects fibred knots

By Yi Ni. 2007

280 citations recorded by the source corpus.

Ozsvath and Szabo conjectured that knot Floer homology detects fibred knots in S^3. We will prove this conjecture for null-homologous knots in arbitrary closed 3-manifolds. Namely, if K is a knot in a closed 3-manifold Y, Y-K is irreducible, and \hat{HFK}(Y,K) is monic, then K is fibred. The proof relies on previous works due to Gabai, Ozsvath–Szabo, Ghiggini and the author. A corollary is that if a knot in S^3 admits a lens space surgery, then the knot is fibred.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/46da3256b3fc1bcf64c81a72917c7d4de43d4ecb

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Citations in the corpus

Knot Floer homology obstructs ribbon concordance (2019, methodology): “Note that our Theorem 1.1 gives an alternate proof of this fact which avoids Kochloukova’s result, by using Ni’s theorem that knot Floer homology detects fibered knots [Ni07] together with [Gor81, Lemma 3.4].”
On Knot Floer Homology and Cabling: 2 (2008, background): “In particular, a corollary of their theorem is that the coefficients of the Alexander polynomial of an L-space knot are all ±1, and that the knot must be fibered [5, 20] with four-genus equal to the Seifert genus [24].”
On the Khovanov and knot Floer homologies of quasi-alternating links (2007, background): “Knot Floer homology detects the genus of a knot [17], as well as whether a knot is fibered [25].”
A survey on Heegaard Floer homology and concordance (2015, background): “For example, Heegaard Floer homology detects fiberedness of knots and three-manifolds [Ghi08, Ni07, Ni09], and bounds the four-ball genus and unknotting number [OS03b, OS05a].”
Instanton Floer homology and the Alexander polynomial (2009, methodology): “In [8], the authors adapted the argument of Ni [11] to establish a criterion for a knot K in S3 to be a fibered knot: in particular, Corollary 7.”
Grid Homology for Knots and Links (2015, result): “The converse to this result, proved by Paolo Ghiggini in the case where g(K) = 1 [71] and in general by Yi Ni [160], is much deeper and gives the following: Theorem 1.”
On the geography and botany of knot Floer homology (2014, background): “2] K is fibered Ghiggini [22], Ni [59] K is strongly quasipositive [28, Proposition 2.”
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].”
Cabling, contact structures and mapping class monoids (2010, background): “In particular, any knot in S3 with a Dehn surgery yielding a lens space is known to be fibered [48], [45].”
Floer simple manifolds and L-space intervals (2015, unknown): “More generally, it is known that ĤFK(K) determines both the Thurston norm of Y and whether it is fibered [36,35,27].”
Contact structures, sutured Floer homology and TQFT (2008, methodology): “Knot Floer homology is a powerful invariant which detects the genus of a knot by the work of Ozsv´ ath-Szab´ o [OS6], and detects fibered knots by the work of Ghiggini [Gh] and Ni [ Ni ].”
A survey of Heegaard Floer homology (2013, methodology): “The genus one case was proved by Ghiggini [22], and the general case was proved simultaneously by Ni [60, 61] and the author [35, 38].”
Ordered Groups and Topology (2015, background): “By Yi Ni [80] if surgery on K yields an L-space, K must be fibred.”