Legendrian and Transversal Knots

By John B. Etnyre. 2003

286 citations recorded by the source corpus.

Publisher Summary Contact structures on manifolds and Legendrian and transversal knots in them are very natural objects, born over a century ago, in the work of Huygens, Hamilton, and Jacobi on geometric optics and work of Lie on partial differential equations. This chapter discusses Legendrian and transversal knots in dimension three where their theory is most fully developed and where they are most intimately tied to topology. In this dimension a predominately topological and combinatorial approach may be used to their study. The chapter then examines higher dimensional Legendrian knots. The chapter also explains the concept of Lagrangian protection. Analogous to the definition of finite type invariants of topological knots, finite type invariants of Legendrian knots can be defined. It is then possible to get a global contact isotopy if there is a fixed over twisted disk in the complement of both Legendrian knot. A complete understanding of Legendrian unknots in over twisted contact structures must wait for future work.

Subfields: Genetics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ad6bd4aa3220784dbbbff1f6498fb7cd43c3ede7

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Citations in the corpus

Augmentations and rulings of Legendrian knots (2004, unknown): “For a more comprehensive introduction, see [8, 17].”
Grid Homology for Knots and Links (2015, background): “If the integral from 0 to 2π vanishes, the diagram lifts to a closed, immersed curve, which furthermore is embedded if the two zcoordinates over any double points are different; see also [45].”
Topologically Trivial Legendrian Knots (2008, methodology): “We refer the reader to Etnyre’s paper [Et2] for a survey of the status of Legendrian isotopy problem back in 2003.”
Surgery diagrams for contact 3-manifolds (2003, background): “st = xdy ydx+zdt tdz or, equivalently, as the complex tangencies of S3 C2. For other basics of contact geometry we refer to [5]; for Legendrian knots, their presentation via front projections, and their classical invariants tb (Thurston-Bennequin invariant) and rot (rotation number) see [12], [ 8 ];”
Handle moves in contact surgery diagrams (2008, background): “The Legendrian band-sum with such a Legendrian unknot is trivial, provided the band is disjoint from a disc bounded by L0 : the Legendrian isotopy between L#LbL0 and the original Legendrian knot L is given, in Darboux charts, by a sequence of first and second Legendrian Reidemeister moves (see [33] ”
Legendrian fronts for affine varieties (2016, methodology): “In contact topology, it is often the case that an immersed point in the Lagrangian projection is said to correspond to a Reeb chord in the Legendrian front projection [32], but the length of the Reeb chord actually depends on a choice of potential.”
Transverse knots and Khovanov homology (2004, methodology): “We collect a few necessary facts about transverse knots and their relation to braids here, referring the reader to [ Et ] for a detailed survey.”
Right-veering diffeomorphisms of compact surfaces with boundary (2005, background): “A good reference for invariants of transverse and Legendrian knots is [ Et ].”
Rational linking and contact geometry (2009, background): “In Section 6, we briefly discuss the generalization of our results to the case of links.”
Transverse knots distinguished by knot Floer homology (2007, background): “Legendrian and transverse knots play a central role in contact geometry; see Etnyre’s survey [12] for further background.”
Fibered Transverse Knots and the Bennequin Bound (2008, unknown): “07 58 v3 [ m at h. SG ] 9 J ul 2 00”
First steps in stable Hamiltonian topology (2010, unknown): “[24]), we call an oriented knot γ : S(1) → M Legendrian if λ(γ̇) ≡ 0 and (positively) transverse if λ(γ̇) > 0.”
Obstructions to Lagrangian concordance (2014, background): “We assume basic familiarity with the theory of Legendrian knots, along the lines of [18].”