Bordered Floer homology and a meridional class of knot

Using a Heegaard diagram for the pullback of a knot K ⊂ S3 in its double branched cover Σ2(K), we give a combinatorial proof for the invariance of the associated knot Floer homology over ℤ.

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Knot doubling operators and bordered Heegaard Floer homology

Journal of Topology ◽

10.1112/jtopol/jts021 ◽

2012 ◽

Vol 5 (3) ◽

pp. 651-712 ◽

Cited By ~ 7

Author(s):

Adam Simon Levine

Keyword(s):

Floer Homology ◽

Heegaard Floer Homology ◽

Heegaard Floer

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Heegaard Floer homology as morphism spaces

Quantum Topology ◽

10.4171/qt/25 ◽

2011 ◽

pp. 381-449 ◽

Cited By ~ 6

Author(s):

Robert Lipshitz ◽

Peter Ozsváth ◽

Dylan Thurston

Keyword(s):

Floer Homology ◽

Heegaard Floer Homology ◽

Heegaard Floer

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The sutured Floer homology polytope

Geometry & Topology ◽

10.2140/gt.2010.14.1303 ◽

2010 ◽

Vol 14 (3) ◽

pp. 1303-1354 ◽

Cited By ~ 15

Author(s):

András Juhász

Keyword(s):

Floer Homology

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The Arnold conjecture, Floer homology and symplectic homology

Symplectic Invariants and Hamiltonian Dynamics ◽

10.1007/978-3-0348-8540-9_6 ◽

1994 ◽

pp. 193-271

Author(s):

Helmut Hofer ◽

Eduard Zehnder

Keyword(s):

Floer Homology ◽

Symplectic Homology ◽

Arnold Conjecture

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Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree

Journal of Topology ◽

10.1112/jtopol/jtw003 ◽

2016 ◽

Vol 9 (2) ◽

pp. 607-686

Author(s):

Robert Lipshitz ◽

Peter S. Ozsváth ◽

Dylan P. Thurston

Keyword(s):

Spectral Sequence ◽

Floer Homology ◽

Double Cover ◽

Spectral Sequences

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ESTIMATES AND COMPUTATIONS IN RABINOWITZ–FLOER HOMOLOGY

Journal of Topology and Analysis ◽

10.1142/s1793525309000205 ◽

2009 ◽

Vol 01 (04) ◽

pp. 307-405 ◽

Cited By ~ 22

Author(s):

ALBERTO ABBONDANDOLO ◽

MATTHIAS SCHWARZ

Keyword(s):

Maximum Principle ◽

Energy Level ◽

Exact Sequence ◽

Floer Homology ◽

Energy Functional ◽

Convex Compact ◽

Closed Geodesics ◽

Action Functional ◽

Uniform Estimates ◽

Rabinowitz Floer Homology

The Rabinowitz–Floer homology of a Liouville domain W is the Floer homology of the Rabinowitz free period Hamiltonian action functional associated to a Hamiltonian whose zero energy level is the boundary of W. This invariant has been introduced by K. Cieliebak and U. Frauenfelder and has already found several applications in symplectic topology and in Hamiltonian dynamics. Together with A. Oancea, the same authors have recently computed the Rabinowitz–Floer homology of the cotangent disk bundle D* M of a closed Riemannian manifold M, by means of an exact sequence relating the Rabinowitz–Floer homology of D* M with its symplectic homology and cohomology. The first aim of this paper is to present a chain level construction of this exact sequence. In fact, we show that this sequence is the long homology sequence induced by a short exact sequence of chain complexes, which involves the Morse chain complex and the Morse differential complex of the energy functional for closed geodesics on M. These chain maps are defined by considering spaces of solutions of the Rabinowitz–Floer equation on half-cylinders, with suitable boundary conditions which couple them with the negative gradient flow of the geodesic energy functional. The second aim is to generalize this construction to the case of a fiberwise uniformly convex compact subset W of T* M whose interior part contains a Lagrangian graph. Equivalently, W is the energy sublevel associated to an arbitrary Tonelli Lagrangian L on TM and to any energy level which is larger than the strict Mañé critical value of L. In this case, the energy functional for closed geodesics is replaced by the free period Lagrangian action functional associated to a suitable calibration of L. An important issue in our analysis is to extend the uniform estimates for the solutions of the Rabinowitz–Floer equation — both on cylinders and on half-cylinders — to Hamiltonians which have quadratic growth in the momenta. These uniform estimates are obtained by the Aleksandrov integral version of the maximum principle. In the case of half-cylinders, they are obtained by an Aleksandrov-type maximum principle with Neumann conditions on part of the boundary.

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