Journal ArticleJournal of Topology · December 1, 2022
We present the first examples of elements in the fundamental group of the space of Legendrian links in (Formula presented.) whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links ...
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Journal ArticleAdvances in Theoretical and Mathematical Physics · January 1, 2020
We sketch a construction of Legendrian Symplectic Field Theory (SFT) for conormal tori of knots and links. Using large N duality and Witten’s connection between open Gromov–Witten invariants and Chern–Simons gauge theory, we relate the SFT of a link conorm ...
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Journal ArticleInventiones Mathematicae · March 1, 2018
We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement cons ...
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Journal ArticleJournal de l'Ecole Polytechnique - Mathematiques · January 1, 2017
The conormal Lagrangian LK of a knot K in R3 is the submanifold of the cotangent bundle T∗R3 consisting of covectors along K that annihilate tangent vectors to K. By intersecting with the unit cotangent bundle S∗R3, one obtains the unit conormal ΛK, and th ...
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Journal ArticleQuantum Topology · September 9, 2015
In [31], O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two re_nements of Plamenevskaya’s invariant, one valued in Bar-Natan’s deformation (from [2]) of the Khovanov complex and anothe ...
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Journal ArticleJournal of Differential Geometry · September 1, 2015
We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in S1 × S2 or any connected sum #k(S1 ×S2), viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-bal ...
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Journal ArticleAdvances in Theoretical and Mathematical Physics · January 1, 2014
We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov- Witten disk amplitudes of a Lagrangian associated to a knot ...
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Journal ArticleAlgebraic and Geometric Topology · August 1, 2013
We develop a close relation between satellites of Legendrian knots in ℝ3 and the Chekanov-Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in ℝ3 and augmenta ...
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Journal ArticleGeometry and Topology · May 13, 2013
The conormal lift of a link K in ℝ3 is a Legendrian submanifold ∧K in the unit cotangent bundle U*ℝ3 of ℝ3 with contact structure equal to the kernel of the Liouville form. Knot contact homology, a topological link invariant of K, is defined as the Legendr ...
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Journal ArticleMathematische Annalen · April 1, 2013
We construct a new invariant of transverse links in the standard contact structure on ℝ3 This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link, see (Ekholm et al., Knot contact homology, Arxi ...
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Journal ArticleAdvances in Mathematics · August 20, 2011
We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and produces a three-v ...
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Journal ArticleInventiones Mathematicae · December 1, 2010
We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary number of positive ...
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Journal ArticleAlgebraic and Geometric Topology · July 28, 2010
We apply knot Floer homology to exhibit an infinite family of transversely nonsimple prime knots starting with 10132. We also discuss the combinatorial relationship between grid diagrams, braids and Legendrian and transverse knots in standard contact R3. ...
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Journal ArticleProceedings of Gokova Geometry-Topology Conference 2008, 120--136 · December 19, 2008
We use grid diagrams to present a unified picture of braids, Legendrian
knots, and transverse knots. ...
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Journal ArticleInternational Mathematics Research Notices · December 1, 2008
We give a simple unified proof for several disparate bounds on Thurston-Bennequin number for Legendrian knots and self-linking number for transverse knots in 3, and provide a template for possible future bounds. As an application, we give sufficient condit ...
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Journal ArticleDuke Mathematical Journal · February 1, 2008
We extend knot contact homology to a theory over the ring ℤ[λ±1, μ±1] with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in S3 and can be generalized to knots in arbitrary manifolds, distin ...
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Journal ArticlePacific Journal of Mathematics · March 1, 2006
We strengthen the link between holomorphic and generating-function invariants of Legendrian knots by establishing a formula relating the number of augmentations of a knot's contact homology to the complete ruling invariant of Chekanov and Pushkar. ...
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Journal ArticleGifted Child Quarterly · January 1, 2006
If the academic needs of the most profoundly gifted students can be met through the use of existing educational practices, specialists in gifted education can assume that the educational needs of less able, but still academically talented, students can als ...
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Journal ArticleGeometry and Topology · August 26, 2005
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 k ...
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Journal ArticleProceedings of G\~A{\P}kova Geometry-Topology Conference 2005
(2006), 165-174 · March 8, 2005
We apply contact homology to obtain new results in the problem of
distinguishing immersed plane curves without dangerous self-tangencies. ...
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Journal ArticleGeometry and Topology · January 26, 2005
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 ...
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Journal ArticleTopology · January 1, 2003
We establish tools to facilitate the computation and application of the Chekanov-Eliashberg differential graded algebra (DGA), a Legendrian-isotopy invariant of Legendrian knots and links in standard contact three space. More specifically, we reformulate t ...
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Journal ArticleAppears as part of"Legendrian solid-torus links", J. Symplectic
Geom. 2 (2005), No. 3, 411-443 · December 11, 2001
We examine the Legendrian analogue of the topological satellite construction
for knots, and deduce some results for specific Legendrian knots and links in
standard contact three-space and the solid torus. In particular, we show that
the Chekanov-Eliashberg ...
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Journal Article · August 28, 2000
We resolve a question of Fuchs and Tabachnikov by showing that there is a
Legendrian knot in standard contact three-space with zero Maslov number which
is not Legendrian isotopic to its mirror. The proof uses the differential
graded algebras of Chekanov. ...
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Journal ArticleJournal of Graph Theory · January 1, 1997
A hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v1, v2, . . . , vk of k distinct vertices of G, there exists a hamiltonian cycle that encounters v1, v2, . . . , vk in this order. Theorems by Dirac and Ore, presenting suffici ...
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