Legendrian and Transversal Knots
This is a survey paper on Legendrian and transversal knots for Handbook of Knot Theory.
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arXiv subjects
John B. Etnyre
Publications and source records attributed to John B. Etnyre.
Planar open book decompositions and contact structures
In this note we observe that while all overtwisted contact structures on compact 3--manifolds are supported by planar open book decompositions, not all contact structures are. This has relevance to invariants of contact structures and also to the Weinstein conjecture via work of Abbas Cieliebak and Hofer.
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Invariants of Legendrian Knots and Coherent Orientations
We provide a translation between Chekanov's combinatorial theory for invariants of Legendrian knots in the standard contact R^3 and a relative version of Eliashberg and Hofer's Contact Homology. We use this translation to transport the idea of ``coherent orientations'' from the Contact Homology world to Chekanov's combinatorial setting. As a result, we obtain a lifting of Chekanov's differential graded algebra invariant to an algebra over Z[t,t^{-1}] with a full Z grading.
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Generic hydrodynamic instability of curl eigenfields
We prove that for generic geometry, the curl-eigenfield solutions to the steady Euler equations on the three torus are all hydrodynamically unstable (linear, L^2 norm). The proof involves a marriage of contact topological methods with the instability criterion of Friedlander-Vishik. An application of contact homology is the crucial step.
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Introductory Lectures on Contact Geometry
These notes are an expanded version of an introductory lecture on contact geometry given at the 2001 Georgia Topology Conference. They are intended to present some of the "topological" aspects of three dimensional contact geometry.
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Knots and Contact Geometry II: Connected Sums
We study the behavior of Legendrian and transverse knots under the operation of connected sums. As a consequence we show that there exist Legendrian knots that are not distinguished by any known invariant. Moreover, we classify Legendrian knots in some non-Legendrian simple knot types.
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On Symplectic Cobordisms
In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3-manifolds are ``symplectic cobordism equivalent.''
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Tight contact structures with no symplectic fillings
We exhibit tight contact structures on 3-manifolds that do not admit any symplectic fillings.
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Knots and Contact Geometry
We classify Legendrian torus knots and figure eight knots in the tight contact structure on the 3-sphere up to Legendrian isotopy. As a corollary to this we also obtain the classification of transversal torus knots and figure eight knots up to transversal isotopy.
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On the non-existence of tight contact structures
We exhibit a 3-manifold which admits no tight contact structure.
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Transversal torus knots
We classify positive transversal torus knots in tight contact structures up to transversal isotopy.
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Stratified integrals and unknots in invisid flows
We prove that any steady solution to the real analytic Euler equations on a Riemannian 3-sphere must possess a periodic orbit bounding an embedded disc. One key ingredient is an extension of Fomenko's work on the topology of integrable Hamiltonian systems to a degenerate case involving stratified integrals. The result on the Euler equations follows from this when combined with some contact-topological perspectives and a recent result of Hofer, Wyzsocki, and Zehnder.
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