Contact homology and one parameter families of Legendrian knots

Kálmán, Tamás

doi:10.2140/gt.2005.9.2013

Cited by 27 publications

(15 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following results from [15] allow us to modify a sequence of Lagrangian Reidemeister moves by removing canceling pairs of moves and commuting pairs of moves that are far away from each other. These modifications do not change the resulting bijection on augmentation chain homotopy classes.…”

Section: Definingmentioning

confidence: 99%

Connections between Floer-type invariants and Morse-type invariants of Legendrian knots

Henry¹

2011

Pacific J. Math.

View full text Add to dashboard Cite

We define an algebraic/combinatorial object on the front projection of a Legendrian knot, called a Morse complex sequence, abbreviated MCS. This object is motivated by the theory of generating families and provides new connections between generating families, normal rulings, and augmentations of the Chekanov-Eliashberg DGA. In particular, we place an equivalence relation on the set of MCSs on and construct a surjective map from the equivalence classes to the set of chain homotopy classes of augmentations of L , where L is the Ng resolution of . In the case of Legendrian knot classes admitting representatives with two-bridge front projections, this map is bijective. We also exhibit two standard forms for MCSs and give explicit algorithms for finding these forms. The definition of an MCS, the equivalence relation, and the statements of some of the results originate from unpublished work of Petya Pushkar.

show abstract

Section: Definingmentioning

confidence: 99%

Connections between Floer-type invariants and Morse-type invariants of Legendrian knots

Henry¹

2011

Pacific J. Math.

View full text Add to dashboard Cite

show abstract

“…The resulting theory gives a rich set of invariants (see e.g. [5,14,10,12,23,24,21]). The aim of the present paper is to provide a generalization of the second level of SFT to the relative case of exact cobordisms with cylindrical ends.…”

Section: Mathematics Subject Classification (2000): 57r17 53d12 57rmentioning

confidence: 99%

Rational symplectic field theory over $\mathbb{Z}_2$ for exact Lagrangian cobordisms

Ekholm¹

2008

J. Eur. Math. Soc.

135

View full text Add to dashboard Cite

We construct a version of rational symplectic field theory for pairs (X, L), where X is an exact symplectic manifold, where L ⊂ X is an exact Lagrangian submanifold with components subdivided into k subsets, and where both X and L have cylindrical ends. The theory associates to (X, L) a Z-graded chain complex of vector spaces over Z 2 , filtered with k filtration levels. The corresponding k-level spectral sequence is invariant under deformations of (X, L) and has the following property: if (X, L) is obtained by joining a negative end of a pair (X , L) to a positive end of a pair (X , L), then there are natural morphisms from the spectral sequences of (X , L) and of (X , L) to the spectral sequence of (X, L). As an application, we show that if ⊂ Y is a Legendrian submanifold of a contact manifold then the spectral sequences associated to (Y × R, s k × R), where Y × R is the symplectization of Y and where s k ⊂ Y is the Legendrian submanifold consisting of s parallel copies of subdivided into k subsets, give Legendrian isotopy invariants of .

show abstract

“…For contact homology of Legendrian knots in R 3 , Kálmán [24] constructed a combinatorial version of this monodromy (which is equivalent to the monodromy defined analytically by counting holomorphic discs [11]), and used it to detect a homotopically nontrivial one-parameter family of Legendrian knots in R 3 which is nullhomotopic in the space of smooth embeddings. Example 1.9 (Homotopy groups) Suppose B D S k with k > 1.…”

Section: Invariants Of Familiesmentioning

confidence: 99%

Floer homology of families I

Hutchings

2008

Algebr. Geom. Topol.

View full text Add to dashboard Cite

In principle, Floer theory can be extended to define homotopy invariants of families of equivalent objects (eg Hamiltonian isotopic symplectomorphisms, 3-manifolds, Legendrian knots, etc.) parametrized by a smooth manifold B . The invariant of a family consists of a filtered chain homotopy type, which gives rise to a spectral sequence whose E 2 term is the homology of B with local coefficients in the Floer homology of the fibers. This filtered chain homotopy type also gives rise to a "family Floer homology" to which the spectral sequence converges. For any particular version of Floer theory, some analysis needs to be carried out in order to turn this principle into a theorem. This paper constructs the invariant in detail for the model case of finite dimensional Morse homology, and shows that it recovers the Leray-Serre spectral sequence of a smooth fiber bundle. We also generalize from Morse homology to Novikov homology, which involves some additional subtleties. 57R58Floer theory is a certain kind of generalization of Morse theory, of which there are now a number of different flavors. These give invariants of symplectomorphisms, 3-manifolds, Legendrian knots, and many other types of objects. This paper describes a fundamental structure which apparently exists in most or all versions of Floer theory. The structure in question is a homotopy invariant of families of equivalent objects parametrized by a smooth manifold B . Its different manifestations thus give invariants of families of Hamiltonian isotopic symplectomorphisms, families of 3-manifolds, etc. The invariant of a family consists of a filtered chain homotopy type, which gives rise to a spectral sequence whose E 2 term is the homology of B with local coefficients in the Floer homology of the fibers. This filtered chain homotopy type also gives rise to a "family Floer homology" to which the spectral sequence converges.The general properties of this family invariant are stated in the "Main Principle" below. This principle cannot be formulated as a general theorem, because there is no precise definition of "Floer theory" that encompasses all of its diverse variants. For any particular version of Floer theory, in order to turn the principle into a theorem, one needs to slightly extend the construction of the Floer theory in question and check that the requisite analysis goes through. The rest of this paper constructs the invariant in detail for the model version of Floer theory, namely finite-dimensional Morse homology, in language designed to carry over to other versions of Floer theory. In this model situation, a family consists a smooth fiber bundle whose fibers are closed manifolds, with a family of (generically Morse) functions of the fibers. Here it turns out that the family invariant recovers the Leray-Serre spectral sequence of the fiber bundle.The outline of this paper is as follows. The Main Principle is enunciated in Section 1. Section 2 reviews some aspects of the Morse complex that will be needed here. The spectral sequence and family Floer homol...

show abstract

Contact homology and one parameter families of Legendrian knots

Cited by 27 publications

References 21 publications

Connections between Floer-type invariants and Morse-type invariants of Legendrian knots

Connections between Floer-type invariants and Morse-type invariants of Legendrian knots

Rational symplectic field theory over $\mathbb{Z}_2$ for exact Lagrangian cobordisms

Floer homology of families I

Contact Info

Product

Resources

About