Sutured Floer homology distinguishes between Seifert surfaces

Altman, Irida

doi:10.1016/j.topol.2012.06.002

Cited by 11 publications

(21 citation statements)

References 15 publications

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“…To assign a concrete group HF˝pY, p, sq to the based Spin c 3-manifold pY, p, sq, one needs more. The first steps toward naturality were made by Ozsváth and Szabó [79], and completely worked out by Thurston and the author [39] We would like to warn the reader of the widespread practice of using the word "canonical" for any well-defined map, without checking property (2). Given such a transitive system, we obtain HF˝pY, pq by taking the product of all the groups HF˝pHq, where H is an admissible diagram of pY, pq (note that these form a set as Σ is a subset of Y ), and take elements x in this product such that F s H,H 1 pxpHqq " xpH 1 q for any pair H, H 1 .…”

Section: Invariance and Naturalitymentioning

confidence: 99%

“…This idea was presented by Hedden, Sarkar, and the author [27]. The first example of Seifert surfaces R and R 1 where the graded groups themselves are different was given by Altman [2]. Sutured Floer homology currently has extensions in two different directions.…”

Section: Usingmentioning

confidence: 99%

“…. ÝÑ HF´pY, sq Furthermore, we have an exact sequence (2) . There is one more piece of algebraic structure on HF˝pY, sq, which is an action of the group Λ˚pH 1 pY q{Torsq.…”

mentioning

confidence: 99%

“…where p and i are the maps in the exact sequence (2), while rγs is the class of the curve γ in H 1 pY q{Tors. This map is non-trivial for example when Y " Σp2, 3, 7q#pS 1ˆS2 q and γ " S 1ˆt ptu, the basepoint being an arbitrary element of γ. Heegaard Floer homology enjoys various symmetry properties.…”

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confidence: 99%

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A Survey of Heegaard Floer Homology

Juhász¹

2015

New Ideas in Low Dimensional Topology

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Since its inception in 2001, Heegaard Floer homology has developed into such a large area of low-dimensional topology that it has become impossible to overview all of its applications and ramifications in a single paper. For the state of affairs in 2004, see the excellent survey article of Ozsváth and Szabó [70]. A decade later, this work has two goals. The first is to provide a conceptual introduction to the theory for graduate students and interested researchers, the second is to survey the current state of the field, without aiming for completeness.After reviewing the structure of Heegaard Floer homology, treating it as a black box, we list some of its most important applications. Many of these are purely topological results, not referring to Heegaard Floer homology itself. Then, we briefly outline the construction of Lagrangian intersection Floer homology, as defined by Fukaya, Oh, Ono, and Ohta [16]. Given a strongly s-admissible based Heegaard diagram pΣ, α, β, zq of the Spin c 3-manifold pY, sq, we construct the Heegaard Floer chain complex CF 8 pΣ, α, β, z, sq as a special case of the above, and try to motivate the role of the various seemingly ad hoc features such as admissibility, the choice of basepoint, and Spin c -structures. We also discuss the proof of invariance of the homology HF 8 pΣ, α, β, sq up to isomorphism under all the choices made, and how to define HF 8 pY, sq using this in a functorial way (naturality). Next, we explain why Heegaard Floer homology is computable, and how it lends itself to the various combinatorial descriptions that most students encounter first during their studies. The last chapter gives an overview of the definition and applications of sutured Floer homology, which includes sketches of some of the key proofs.Throughout, we have tried to collect some of the important open conjectures in the area. For example, a positive answer to two of these would give a new proof of the Poincaré conjecture.Acknowledgement. I would like to thank for their comments on earlier versions of this paper. ι ÝÑ HF 8 pY, sq π ÝÑ HF`pY, sq δ ÝÑ . . . This gives rise to the invariantHFr ed pY, sq " cokerpπqkerpιq " HFŕ ed pY, sq, where the isomorphism is given by the coboundary map δ.

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Section: Invariance and Naturalitymentioning

confidence: 99%

Section: Usingmentioning

confidence: 99%

“…. ÝÑ HF´pY, sq Furthermore, we have an exact sequence (2) . There is one more piece of algebraic structure on HF˝pY, sq, which is an action of the group Λ˚pH 1 pY q{Torsq.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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A Survey of Heegaard Floer Homology

Juhász¹

2015

New Ideas in Low Dimensional Topology

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“…Combining the sutured Floer homology of pS 3 pRq, γq with the Seifert form turns out to be a useful tool in distinguishing different Seifert surfaces(see [HJS08]). Altman in [Alt11] 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).…”

Section: Introductionmentioning

confidence: 99%

Seifert surfaces distinguished by sutured Floer homology but not its Euler characteristic

Vafaee

2015

Topology and its Applications

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In this paper we find a family of knots with trivial Alexander polynomial, and construct two non-isotopic Seifert surfaces for each member in our family. 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. Our examples provide the first use of sutured Floer homology, and not merely its Euler characteristic(a classical torsion), to distinguish Seifert surfaces. Our technique uses a version of Floer homology, called "longitude Floer homology" in a way that enables us to bypass the computations related to the SF H of the complement of a Seifert surface.

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Splittings of knot groups

2014

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Let K be a knot of genus g. If K is fibered, then it is well known that the knot group π(K) splits only over a free group of rank 2g. We show that if K is not fibered, then π(K) splits over non-free groups of arbitrarily large rank. Furthermore, if K is not fibered, then π(K) splits over every free group of rank at least 2g. However, π(K) cannot split over a group of rank less than 2g. The last statement is proved using the recent results of Agol, Przytycki-Wise and Wise.

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Sutured Floer homology distinguishes between Seifert surfaces

Cited by 11 publications

References 15 publications

A Survey of Heegaard Floer Homology

A Survey of Heegaard Floer Homology

Seifert surfaces distinguished by sutured Floer homology but not its Euler characteristic

Splittings of knot groups

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