Floer homology is a novel invariant that arises as an infinitedimensional analogue of finitedimensional morse homology. In particular, the rank of the top term of knot floer homology bounds the topological complexity of the knot complement, in addition to simply detecting fibred knots. R that is spanned by the spin cstructures which support nonzero floer homology groups. We first prove an adjunction inequality and then define a polytope pm. Contact handles, duality, and sutured floer homology. If m,g m,g is a taut surface decomposition, then a natural map projects pm,g onto a face of pm,g. Floer polytope equals the dual of the sutured thurston polytope of. In order to distinguish the surfaces we study the sutured floer homology invariants of the sutured manifolds obtained by cutting the knot complements along the seifert surfaces. Using heegaard floer homology, we construct invariants of manifolds with involutions. The article contains most of the things the author wishes she had known when she started her journey into the world of sutured floer homology. In this paper we extend the theory of sutured floer homology developed by the author in 5, 6. We may then consider sutured floer homology, a powerful invariant of sutured manifolds based on pseudoholomorphic curves.
On some properties of the sutured floer polytope core. Seifert surfaces distinguished by sutured floer homology but. In mathematics, floer homology is a tool for studying symplectic geometry and lowdimensional topology. Problems on lowdimensional topology rims, kyoto university. Bordered sutured floer homology columbia university. Floer homology is typically defined by associating to the object of interest an infinitedimensional manifold and a real valued function on it. On a heegaard floer theory for tangles cambridge repository. Ams transactions of the american mathematical society. Altman in gives an example of using only the sutured floer homology polytope to distinguish two seifert surfaces. Actually, we exhibit an infinite family of knots with pairs of seifert surfaces that can be distinguished by the euler characteristic. The first is to provide a conceptual introduction to heegaard floer homology, the second is to survey the current state of the field, without aiming for completeness. It is worth noting that in order to find example 8. However, there isnt as yet a complete answer on the seibergwitten side.
We first prove an adjunction inequality and then define a polytope p m. Sutured floer homology distinguishes between seifert surfaces. Sutured floer homology, denoted by sfh, is an invariant of balanced sutured manifolds previously defined by the author. We exhibit the first example of a knot k in the threesphere with a pair of minimal genus seifert surfaces r1 and r2 that can be distinguished using t. A user interface to the snappea kernel which runs on mac os x, linux, and windows. In this paper we extend the theory of sutured floer homology developed by the author. Sutured floer homology is an invariant of balanced sutured manifolds introduced by.
I will describe aspects of this program and outline a construction of a new class. In this paper, we extend the theory of sutured floer homology developed by the author. In the symplectic version, this is the free loop space of a symplectic manifold with the symplectic action functional. Maksim lipyanskiy mit a semiinfinite cycle construction of floer homology. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Hmy,s agrees with the sutured monopole floer homology group. On simplification of combinatorial link floer homology. We will also explain the relation to nadlers program for the arborealization of. Apr 09, 20 this article is a standalone introduction to sutured floer homology for graduate students in geometry and topology. Sutured floer homology polytope and decomposition of a sutured manifold along an embedded surface. Moreover, sfh acts as a complexity for balanced sutured manifolds. In this paper, we prove that for a taut balanced sutured manifold with vanishing second homology, the dimension of the.
By applying seiferts algorithm to a special alternating diagram of a link l, one obtains a seifert surface f of l. Sutured instanton floer homology was introduced by kronheimer and mrowka. Research institute for the program homology theories of knots and links. R that is spanned by the spincstructures which support nonzero floer homology groups. In this paper we give a formula that shows how this invariant changes under. Each chapter of this thesis is a selfcontained article on sutured floer homology. For closed 3manifolds, heegaard floer homology is related to the thurston norm through results due to ozsv\ath and szab\o, ni, and hedden. Seifert surfaces distinguished by sutured floer homology.
The third part covers the background on sutured manifolds, the definition of\ud sutured floer homology, as well as a discussion of its most basic properties and implications\ud it detects the product, behaves nicely under surface decompositions, defines an\ud asymmetric polytope, its euler characteristic is computable using fox calculus. For closed 3manifolds, heegaard floer homology is related to the thurston norm through results due to ozsvath and szabo, ni, and hedden. Yanki lekili mit wrinkled fibrations on nearsymplectic manifolds. We first prove an adjunction inequality, and then define a polytope pm,g in h2m,\partial m. Andy manion has already plugged our answer for heegaard floer homology. In this paper we find a family of knots with trivial alexander polynomial, and construct two nonisotopic seifert surfaces for each member in our family.
We first prove an adjunction inequality, and then define a polytope pm,g in h2m,\\partial m. The second part is a construction of heegaard floer homology as a. On the seibergwitten side, not as much is known, but tim nguyens thesis starts to attack the problem. This is a generalization of the dual thurston norm polytope of a linkcomplement studied by ozsvath and szabo using link floer. More precisely, we show that the euler characteristic of the sutured floer homology of the complementary manifolds distinguishes between the two surfaces, as does the sutured floer polytope introduced by juh\asz. Is there a version of seibergwittenfloer or heegard. The decategorification of sutured floer homology uni regensburg. We show that the support of the sutured floer homology of the sutured manifold complementary to f is affine isomorphic to the set of lattice points given as hypertrees in a certain hypergraph that is naturally associated to the diagram. Gt 9 apr 20 the sutured floer polytope and taut depth one foliations irida altman abstract. Yaron ostrover mit on some algebraic properties of the quantum homology. On some properties of the sutured floer polytope wrap.
Bordered sutured floer homology rumen zarev we investigate the relationship between two versions of heegaard floer homology for 3manifolds with boundary the sutured floer homology of juhasz, and the bordered heegaard floer homology of lipshitz, ozsv ath, and thurston. The sutured floer homology polytope nasaads in this paper, we extend the theory of sutured floer homology developed by the author. This article is a standalone introduction to sutured floer homology for graduate students in geometry and topology. Altman in gives an example of using only the sutured floer homology polytope to distinguish two seifert surfaces for a knot for related definitions see section 2. This turns out to contain polytopes as well, and the three sutured floer homology groups associated to a trinity exhibit similar triality relations. We first prove an adjunction inequality, and then define a.
Snappy combines a link editor and 3dgraphics for dirichlet domains and cusp neighborhoods with a powerful commandline interface based on the python programming language. April 11, andras juhasz princeton, the sutured floer homology polytope. Andreas floer introduced the first version of floer homology, now called lagrangian floer homology, in his proof of the arnold conjecture in symplectic geometry. For tangles without closed components, the mathematica program. Sutured floer homology distinguishes between seifert. Our technique uses a version of floer homology, called. We first prove an adjunction inequality, and then define a polytope. Sutured floer homology distinguishes between seifert surfaces authors. We show that there is a bijection between vertices of the sutured floer polytope carrying the group z and equivalence classes of taut depth one foliations that form the foliation cones of cantwell and conlon. The third part covers the background on sutured manifolds, the definition of sutured floer homology, as well as a discussion of its most basic properties and implications it detects the product, behaves nicely under surface decompositions, defines an asymmetric polytope, its euler characteristic is computable using fox calculus. Top kodi archive and support file community software msdos vintage software apk cdrom software cdrom software. Our examples provide the first use of sutured floer homology. The sutured floer polytope and taut depth one foliations. More precisely, we show that the euler characteristic of the sutured floer homology distinguishes between r 1 and r 2, as does the sutured floer polytope introduced by juha.
Columbia symplectic geometry and gauge theory seminar. Dec 12, 2011 sutured floer homology g roups sf h m f, l are completely determined by their euler characteristic according to 5, corolla ry 6. This article is a purely expository introduction to sutured floer homology for graduate students in geometry and topology. For closed 3manifolds, heegaard floer homology is related to the thurston. Apr 09, 20 the third part covers the background on sutured manifolds, the definition of sutured floer homology, as well as a discussion of its most basic properties and implications it detects the product, behaves nicely under surface decompositions, defines an asymmetric polytope, its euler characteristic is computable using fox calculus. Sutured instanton floer homology was introduced by kronheimer and. Many of these are purely topological results, not referring to heegaard floer homology itself.
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