Yaron ostrover mit on some algebraic properties of the quantum homology. Research institute for the program homology theories of knots and links. I will describe aspects of this program and outline a construction of a new class. We first prove an adjunction inequality and then define a polytope pm. We first prove an adjunction inequality, and then define a. Sutured floer homology is an invariant of balanced sutured manifolds introduced by. Columbia symplectic geometry and gauge theory seminar. Many of these are purely topological results, not referring to heegaard floer homology itself. In this paper we extend the theory of sutured floer homology developed by the author in 5, 6. In this paper, we prove that for a taut balanced sutured manifold with vanishing second homology, the dimension of the. R that is spanned by the spin cstructures which support nonzero floer homology groups. Floer homology is typically defined by associating to the object of interest an infinitedimensional manifold and a real valued function on it. This article is a standalone introduction to sutured floer homology for graduate students in geometry and topology.
On the seibergwitten side, not as much is known, but tim nguyens thesis starts to attack the problem. 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. Yanki lekili mit wrinkled fibrations on nearsymplectic manifolds. Problems on lowdimensional topology rims, kyoto university. Is there a version of seibergwittenfloer or heegard. R that is spanned by the spincstructures which support nonzero floer homology groups. 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. This article is a purely expository introduction to sutured floer homology for graduate students in geometry and topology. 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. In this paper we extend the theory of sutured floer homology developed by the author. Floer homology is a novel invariant that arises as an infinitedimensional analogue of finitedimensional morse homology. Top kodi archive and support file community software msdos vintage software apk cdrom software cdrom software. Sutured floer homology polytope and decomposition of a sutured manifold along an embedded surface.
We will also explain the relation to nadlers program for the arborealization of. Sutured floer homology, denoted by sfh, is an invariant of balanced sutured manifolds previously defined by the author. 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. 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. For closed 3manifolds, heegaard floer homology is related to the thurston. On simplification of combinatorial link floer homology. Andreas floer introduced the first version of floer homology, now called lagrangian floer homology, in his proof of the arnold conjecture in symplectic geometry. Sutured instanton floer homology was introduced by kronheimer and. Moreover, sfh acts as a complexity for balanced sutured manifolds. April 11, andras juhasz princeton, the sutured floer homology polytope. However, there isnt as yet a complete answer on the seibergwitten side. If m,g m,g is a taut surface decomposition, then a natural map projects pm,g onto a face of pm,g.
Actually, we exhibit an infinite family of knots with pairs of seifert surfaces that can be distinguished by the euler characteristic. Contact handles, duality, and sutured floer homology. Altman in gives an example of using only the sutured floer homology polytope to distinguish two seifert surfaces. Floer polytope equals the dual of the sutured thurston polytope of. Our technique uses a version of floer homology, called. Sutured instanton floer homology was introduced by kronheimer and mrowka. Seifert surfaces distinguished by sutured floer homology. Sutured floer homology distinguishes between seifert surfaces authors. Seifert surfaces distinguished by sutured floer homology but. Hmy,s agrees with the sutured monopole floer homology group.
In the symplectic version, this is the free loop space of a symplectic manifold with the symplectic action functional. On some properties of the sutured floer polytope core. 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. 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. 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. For closed 3manifolds, heegaard floer homology is related to the thurston norm through results due to ozsvath and szabo, ni, and hedden. Ams transactions of the american mathematical society. Andy manion has already plugged our answer for heegaard floer homology. Sutured floer homology distinguishes between seifert surfaces.
In mathematics, floer homology is a tool for studying symplectic geometry and lowdimensional topology. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Apr 09, 20 this article is a standalone introduction to sutured floer homology for graduate students in geometry and topology. On a heegaard floer theory for tangles cambridge repository. 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. 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. 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. On some properties of the sutured floer polytope wrap. It is worth noting that in order to find example 8. Using heegaard floer homology, we construct invariants of manifolds with involutions. For tangles without closed components, the mathematica program. We first prove an adjunction inequality and then define a polytope p m.
In this paper we give a formula that shows how this invariant changes under. 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. Gt 9 apr 20 the sutured floer polytope and taut depth one foliations irida altman abstract. Maksim lipyanskiy mit a semiinfinite cycle construction of floer homology. 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 is a generalization of the dual thurston norm polytope of a linkcomplement studied by ozsvath and szabo using link floer. A user interface to the snappea kernel which runs on mac os x, linux, and windows. Bordered sutured floer homology columbia university. In this paper, we extend the theory of sutured floer homology developed by the author. We first prove an adjunction inequality, and then define a polytope. Sutured floer homology distinguishes between seifert. We may then consider sutured floer homology, a powerful invariant of sutured manifolds based on pseudoholomorphic curves. We first prove an adjunction inequality, and then define a polytope pm,g in h2m,\\partial m.
This turns out to contain polytopes as well, and the three sutured floer homology groups associated to a trinity exhibit similar triality relations. Snappy combines a link editor and 3dgraphics for dirichlet domains and cusp neighborhoods with a powerful commandline interface based on the python programming language. 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. We first prove an adjunction inequality, and then define a polytope pm,g in h2m,\partial m. Our examples provide the first use of sutured floer homology. 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. By applying seiferts algorithm to a special alternating diagram of a link l, one obtains a seifert surface f of l. The sutured floer homology polytope nasaads in this paper, we extend the theory of sutured floer homology developed by the author. The second part is a construction of heegaard floer homology as a.