The contact homology of Legendrian submanifolds in R2n+1

Ekholm, Tobias; Etnyre, John B.; Sullivan, Michael C.

doi:10.4310/jdg/1143651770

Cited by 120 publications

(338 citation statements)

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“…There is also an obvious relation between Lagrangian concordance and Legendrian contact homology. Following Bourgeois [5] we see that a Lagrangian cylinder between two Legendrian knots could be used to define a map between the algebras CH .K C / and CH .K / (see Ekholm, Etnyre and Sullivan [6]). We, however, will not give a more detailed description of this map for two reasons.…”

Section: Discussionmentioning

confidence: 99%

Lagrangian concordance of Legendrian knots

Chantraine

2010

Algebr. Geom. Topol.

116

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In this article we define Lagrangian concordance of Legendrian knots, the analogue of smooth concordance of knots in the Legendrian category. In particular we study the relation of Lagrangian concordance under Legendrian isotopy. The focus is primarily on the algebraic aspects of the problem. We study the behavior of the classical invariants under this relation, namely the Thurston-Bennequin number and the rotation number, and we provide some examples of non-trivial Legendrian knots bounding Lagrangian surfaces in $D^4$. Using these examples, we are able to provide a new proof of the local Thom conjecture.Comment: 18 pages, 4 figures. v2: Minor corrections and a proof of conjecture 6.4 of version 1 (now Theorem 6.4). v3: Several substantial changes notably the proof of theorems 1.2 and 5.1, this is the version accepted for publication in "Algebraic and Geometric Topology" published in January 201

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Section: Discussionmentioning

confidence: 99%

Lagrangian concordance of Legendrian knots

Chantraine

2010

Algebr. Geom. Topol.

116

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“…This should clearly be related to the theory of Legendrian contact homology, which was described informally by Eliashberg, Givental and Hofer [5, §2.8], and by Chekanov [4] for Legendrian knots in R 3 , and has been developed rigorously by Ekholm, Etnyre and Sullivan [6,7], for embedded Legendrians L in R 2n+1 and in M × R for (M, ω) an exact symplectic manifold. In particular, for (M, ω) exact one can compare our HF * (L, b; Λ Z nov ) for embedded Legendrians in M × S 1 , and Ekholm et al's HC * (L, J) for embedded Legendrians L in M × R, [7].…”

Section: Bounding Cochains and Lagrangian Floer Cohomologymentioning

confidence: 99%

Immersed Lagrangian floer theory

Akaho¹,

Joyce²

2010

J. Differential Geom.

188

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Abstract. Let (M, ω) be a compact symplectic 2n-manifold, and L a compact embedded Lagrangian submanifold in M . Fukaya, Oh, Ohta and Ono [9] construct Lagrangian Floer cohomology for such M, L, yielding groups HF * (L, b; Λnov) for one Lagrangian or HF * `( L 1 , b 1 ), (L 2 , b 2 ); Λnov´for two, where b, b 1 , b 2 are choices of bounding cochains, and exist if and only if L, L 1 , L 2 have unobstructed Floer cohomology. These are independent of choices up to canonical isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M, ω), and so are an essential part of the Homological Mirror Symmetry Conjecture of Kontsevich.The goal of this paper is to extend [9] to immersed Lagrangians L in M with immersion ι : L → M , with transverse self-intersections. In the embedded case, Floer cohomology HF * (L, b; Λnov) is a modified, 'quantized' version of singular homology H n− * (L; Λnov) over the Novikov ring Λnov. In our immersed case, HF * (L, b; Λnov) turns out to be a quantized version ofThe theory becomes simpler and more powerful for graded Lagrangians in Calabi-Yau manifolds, when we can work over a smaller Novikov ring ΛCY. The proofs involve associating a gapped filtered A∞ algebra over Λ 0 nov or Λ 0 CY to ι : L → M , which is independent of nearly all choices up to canonical homotopy equivalence, and is built using a series of finite approximations called A N,0 algebras for N = 0, 1, 2, . . ..

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“…Holomorphic curve techniques stemming from the introduction of Symplectic Field Theory [7] have inspired a great deal of work on Legendrian isotopy invariants; papers in the subject include [3,5,6,8,9,18,23], among others. Most of these papers focus on Legendrian knots, and all deduce results in symplectic and contact geometry.…”

Section: Introductionmentioning

confidence: 99%

Knot and braid invariants from contact homology I

2005

Geom. Topol.

144

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We introduce topological invariants of knots and braid conjugacy classes, in the form of differential graded algebras, and present an explicit combinatorial formulation for these invariants. The algebras conjecturally give the relative contact homology of certain Legendrian tori in five-dimensional contact manifolds. We present several computations and derive a relation between the knot invariant and the determinant.

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The contact homology of Legendrian submanifolds in R2n+1

Cited by 120 publications

References 0 publications

Lagrangian concordance of Legendrian knots

Lagrangian concordance of Legendrian knots

Immersed Lagrangian floer theory

Knot and braid invariants from contact homology I

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