Floer homology and surface decompositions

By Andr'as Juh'asz. 2006

167 citations recorded by the source corpus.

Sutured Floer homology, denoted by SFH, is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if (M, \gamma)--> (M', \gamma') is a sutured manifold decomposition then SFH(M',\gamma') is a direct summand of SFH(M, \gamma). To prove the decomposition formula we give an algorithm that computes SFH(M,\gamma) from a balanced diagram defining (M,\gamma) that generalizes the algorithm of Sarkar and Wang. As a corollary we obtain that if (M, \gamma) is taut then SFH(M,\gamma) is non-zero. Other applications include simple proofs of a result of Ozsvath and Szabo that link Floer homology detects the Thurston norm, and a theorem of Ni that knot Floer homology detects fibred knots. Our proofs do not make use of any contact geometry. Moreover, using these methods we show that if K is a genus g knot in a rational homology 3-sphere Y whose Alexander polynomial has leading coefficient a_g non-zero and if the rank of \hat{HFK}(Y,K,g) < 4 then the knot complement admits a depth < 2 taut foliation transversal to the boundary of N(K).

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/29fd902d88124a70e6dc8bdc785ef9a001d587e6

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Citations in the corpus

The contact invariant in sutured Floer homology (2007, unknown): “In [Ju2], Juhász proved the following theorem:”
Sutures and contact homology I (2010, methodology): “Juhász’ work also motivated the definition of a sutured version of SeibergWitten Floer homology by Kronheimer and Mrowka [KM].”
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 fibred [35, 34, 26].”
Bordered Heegaard Floer homology (2008, methodology): “In a di erent direction, recall that a di erent construction for manifolds-with-boundary, equipped with a suture on the boundary, has been given by András Juhász [15].”
On the geography and botany of knot Floer homology (2014, background): “n knot Floer homology provides about Seifert surfaces is considerably stronger: Genus Detection ([74, Theorem 1.2]). g(K) = max{a ∈ Z | HFK\(K,a) 6= 0 } Fibered Knot Detection ([60, Theorem 1.1], cf. [23, 39]). rank HFK\(K,g(K)) = 1 if and only if K is ﬁbered. Both theorems have extensions to knots ”
The decategorification of sutured Floer homology (2009, background): “Balanced sutured manifolds were defined in [15] and strongly balanced sutured manifolds in [16].”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “An alternative proof was given by Juh´asz [Juh08], using sutured Floer homology.”
Khovanov homology is an unknot-detector (2010, methodology): “The construction of KHI (K) in [24] was motivated by Juhász’s work on sutured manifolds in the setting of Heegaard Floer theory [13, 14]: in the context of sutured manifolds, KHI (K) can be defined as”
On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology (2008, background): “We refer the reader to [8] for the remaining definitions and results about the behavior of sutured Floer homology under surface decompositions.”
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”
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.”