Heegaard–Floer homologies of (+1) surgeries on torus knots

Borodzik, Maciej; Némethi, András

doi:10.1007/s10474-012-0280-x

Cited by 15 publications

(53 citation statements)

References 17 publications

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“…Combining Theorem 2.3 and our Theorem 1.3, it is now easy to calculate the Heegaard-Floer homology of a Seifert homology sphere. Let us illustrate this statement on Y = Σ (2,3,11).…”

Section: Graded Roots and Heegaard-floer Homologymentioning

confidence: 99%

“…Theorem 1.14. A Brieskorn sphere Σ = Σ(p, q, r) is weakly elliptic if and only if (p, q, r) is equal to one of the following triplets: (3,4,5), (2,5,7), (2,5,9), or (2, 3, r) with gcd(6, r) = 1 and r > 5. There are no weakly elliptic Seifert homology spheres with more than three singular fibers.…”

Section: B Can and ç Karakurtmentioning

confidence: 99%

“…However, computing Heegaard-Floer homology for infinite families of 3-manifolds seems to be a formidable combinatorial problem for one has to determine all the local maxima and minima of infinite families of sequences which simultaneously solve an infinite family of non-homogeneous recurrence relations. See [3], [10], and [19] for some particular cases where this problem is handled.…”

Section: Introductionmentioning

confidence: 99%

“…Conjecture 1.5. If an irreducible integral homology sphere Y is an L-space, then Y is either the 3-sphere, or the Poincaré homology sphere Σ (2,3,5) with either orientation.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.6 ([18], [5]). If a Seifert homology sphere Y is an L-space, then Y is homeomorphic to S 3 or ±Σ (2,3,5).…”

Section: Introductionmentioning

confidence: 99%

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Calculating Heegaard–Floer homology by counting lattice points in tetrahedra

Can

Karakurt

2014

Acta Math. Hungar.

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We introduce a notion of complexity for Seifert homology spheres by establishing a correspondence between lattice point counting in tetrahedra and the Heegaard-Floer homology. This complexity turns out to be equivalent to a version of Casson invariant and it is monotone under a natural partial order on the set of Seifert homology spheres. Using this interpretation we prove that there are finitely many Seifert homology spheres with a prescribed Heegaard-Floer homology. As an application, we characterize L-spaces and weakly elliptic manifolds among Seifert homology spheres. Also, we list all the Seifert homology spheres up to complexity two.

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“…Combining Theorem 2.3 and our Theorem 1.3, it is now easy to calculate the Heegaard-Floer homology of a Seifert homology sphere. Let us illustrate this statement on Y = Σ (2,3,11).…”

Section: Graded Roots and Heegaard-floer Homologymentioning

confidence: 99%

Section: B Can and ç Karakurtmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Conjecture 1.5. If an irreducible integral homology sphere Y is an L-space, then Y is either the 3-sphere, or the Poincaré homology sphere Σ (2,3,5) with either orientation.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.6 ([18], [5]). If a Seifert homology sphere Y is an L-space, then Y is homeomorphic to S 3 or ±Σ (2,3,5).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Calculating Heegaard–Floer homology by counting lattice points in tetrahedra

Can

Karakurt

2014

Acta Math. Hungar.

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Heegaard Floer homology and several families of Brieskorn spheres

Tweedy

2013

Topology and its Applications

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In [OS03b], Ozsváth and Szabó gave a combinatorial description for the Heegaard Floer homology of boundaries of certain negative-definite plumbings. Némethi constructed a remarkable algorithm in [Ném05] for executing these computations for almost-rational plumbings, and his work in [Ném07] gives a formula computing the invariants for the Brieskorn homology spheres −Σ (p, q, pqn + 1). Here we give a formula for HF + (−Σ (p, q, pqn − 1)), generalizing the one for the n = 1 case given in [BN11]. We also compute HF + for the families −Σ(2, 5, k) and −Σ(2, 7, k).

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Bordered Floer homology for manifolds with torus boundary via immersed curves

Hanselman¹,

Rasmussen²,

Watson³

2023

J. Amer. Math. Soc.

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This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If M M is such a manifold, we show that the type D structure C F D ^ ( M ) \widehat {CFD}(M) may be viewed as a set of immersed curves decorated with local systems in ∂ M \partial M . These curves-with-decoration are invariants of the underlying three-manifold up to regular homotopy of the curves and isomorphism of the local systems. Given two such manifolds and a homeomorphism h h between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with h h is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of H F ^ \widehat {HF} decreases under a certain class of degree one maps (pinches) and we establish that the existence of an essential separating torus gives rise to a lower bound on the dimension of H F ^ \widehat {HF} . In particular, it follows that a prime rational homology sphere Y Y with H F ^ ( Y ) > 5 \widehat {HF}(Y)>5 must be geometric. Other results include a new proof of Eftekhary’s theorem that L-space homology spheres are atoroidal; a complete characterization of toroidal L-spaces in terms of gluing data; and a proof of a conjecture of Hom, Lidman, and Vafaee on satellite L-space knots.

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Heegaard–Floer homologies of (+1) surgeries on torus knots

Cited by 15 publications

References 17 publications

Calculating Heegaard–Floer homology by counting lattice points in tetrahedra

Calculating Heegaard–Floer homology by counting lattice points in tetrahedra

Heegaard Floer homology and several families of Brieskorn spheres

Bordered Floer homology for manifolds with torus boundary via immersed curves

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