Knot and braid invariants from contact homology I

Ng, Lenhard

doi:10.2140/gt.2005.9.247

Cited by 46 publications

(150 citation statements)

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“…In [7], the author introduced invariants of knots and braid conjugacy classes called knot and braid differential graded algebras (DGAs). The homologies of these DGAs conjecturally give the relative contact homology of certain natural Legendrian tori in 5-dimensional contact manifolds.…”

Section: Resultsmentioning

confidence: 99%

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Knot and braid invariants from contact homology II

2005

Geom. Topol.

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109

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We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant to embedded graphs and higher-dimensional knots. We calculate the knot invariant for two-bridge knots and relate it to double branched covers for general knots.In the appendix we show that the cord ring is determined by the fundamental group and peripheral structure of a knot and give applications.

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Section: Resultsmentioning

confidence: 99%

“…We recall the definitions of degree 0 braid and knot contact homology from [7]. Let A n denote the tensor algebra over Z generated by n(n − 1) generators a ij with 1 ≤ i, j ≤ n, i = j .…”

Section: Background Materialsmentioning

confidence: 99%

Knot and braid invariants from contact homology II

2005

Geom. Topol.

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109

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“…Arguing as above and using the inductive assumption about the kernel of the x @-operator on p 0 ;m 0 . 0 ; / we conclude that (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) ev p .v…”

Section: Building Rigid Flow Treesmentioning

confidence: 85%

“…Equations (6-18) and (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) imply that v Finally, let W OET 1 ; T 2 OE0; 1 ! ‫ރ‬ be a cut off function which is equal to 1 on OE The proofs in the other cases are similar so we just point out the differences from the case just given.…”

Section: Building Rigid Flow Treesmentioning

confidence: 99%

“…The main difference from the case considered is that the kernel of the x @-problem on 0 is smaller. In particular, the analogue of (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) 2-valent puncture r and special punctures p , the special puncture of 0 , and q . In this case replace the problem on OET 1 ; T 2 OE0; 1 considered above by the problem on a 3-punctured disk which is the union of two regions of the form OET 1 ; T 2 OE0; 1, the region B.r /, and OE0; 1/ OE0; 1 corresponding to the 2-valent puncture r .…”

Section: Building Rigid Flow Treesmentioning

confidence: 99%

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Morse flow trees and Legendrian contact homology in 1–jet spaces

Ekholm

2007

Geom. Topol.

259

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1083Morse flow trees and Legendrian contact homology in 1-jet spaces TOBIAS EKHOLMLet L J 1 .M / be a Legendrian submanifold of the 1-jet space of a Riemannian n-manifold M . A correspondence is established between rigid flow trees in M determined by L and boundary punctured rigid pseudo-holomorphic disks in T M , with boundary on the projection of L and asymptotic to the double points of this projection at punctures, provided n Ä 2, or provided n > 2 and the front of L has only cusp edge singularities. This result, in particular, shows how to compute the Legendrian contact homology of L in terms of Morse theory. 57R17; 53D40 IntroductionLet M be a smooth n-manifold, let J 0 .M / D M ‫ޒ‬ be its 0-jet space and let J 1 .M / D T M ‫ޒ‬ be its 1-jet space endowed with the standard contact structure which is the kernel of the 1-form dz p dq , where z is a coordinate in the ‫-ޒ‬direction and where p dq is the canonical 1-form on T M . An n-dimensional submanifold L J 1 .M / is Legendrian if it is everywhere tangent to . This paper concerns Legendrian submanifolds of 1-jet spaces, and in particular their contact homology. Legendrian contact homology is a part of Symplectic Field Theory (see ). It is a framework for finding isotopy invariants of Legendrian submanifolds of contact manifolds by "counting" rigid (pseudo-)holomorphic disks. The analytical foundations of Legendrian contact homology in 1-jet spaces were established in Ekholm-Etnyre-Sullivan [6], see also [4; 5].Contact homology has proved very useful, see eg Chekanov [1], Eliashberg [7] and Ekholm-Etnyre-Sullivan [3], especially for Legendrian submanifolds of dimension 1, where the Riemann mapping theorem can be used to give a combinatorial and computable description of the theory see Chekanov [1] and Etnyre-Ng-Sabloff [10]. In the higher dimensional case, finding holomorphic disks involves solving a non-linear first order partial differential equation. This is an infinite dimensional problem and it is therefore often difficult to compute the contact homology of a given Legendrian A Riemannian metric on M induces an almost complex structure tamed by ! , see Section 4.4.)If S is a Riemann surface with complex structureWe study boundary punctured J -holomorphic disks with boundary mapping to … ‫ރ‬ .L/, which are asymptotic to double points at the punctures, and which have restrictions to the boundary which admit a continuous lift to L. We call such disks J -holomorphic disks with boundary on L. Two of their key properties are as follows. First, the punctures come equipped with signs, see Definition 2.1. Second, associated to a J -holomorphic disk is its formal dimension, see Proposition 3.18, which measures the expected dimension of the space of nearby J -holomorphic disks. We say that a disk is rigid if it has formal dimension 0 and if it is transversely cut out by its defining differential equation. Legendrian contactGeometry & Topology, Volume 11 (2007) Morse flow trees and Legendrian contact homology in 1-jet spaces 1085homology is defined using disks with ...

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