Knots and Contact Geometry

By John B. Etnyre, K. Honda. 2000

62 citations recorded by the source corpus.

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.

Subfields: Computer Graphics and Computer-Aided Design, Geometry and Topology, Mechanical Engineering

View Paper: https://www.semanticscholar.org/paper/51cd5fbe0217082431fa66d3cc4df51da4a17dc8

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In Computer Graphics and Computer-Aided Design

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Citations in the corpus

Heegard Floer invariants of Legendrian knots in contact three-manifolds (2008, background): “(For more background on transverse non-simplicity, see [10, 13].”
Introductory Lectures on Contact Geometry (2001, unknown): “dicate how to use convex surfaces in the study of contact geometry. For applications of this to the classiﬁcation of contact structures see [26] and [21], to the classiﬁcation of Legendrian knots see [14] and to the nature of tightness see [13]. Given a contact manifold (M,ξ) a vector ﬁeld is called”
Invariants of Legendrian Knots and Coherent Orientations (2001, background): “More recently, Etnyre and Honda [ 10 ] proved that torus knots and the figure eight knot are also Legendrian simple.”
Augmentations and rulings of Legendrian knots (2004, background): “Classification results based on these invariants followed in the early 1990’s: Legendrian unknots [6], torus knots [9], and figure eight knots [9] are completely classified by their topological type and classical invariants.”
On the classification of tight contact structures (2002, methodology): “cases used later in Section 4, for a more complete discussion including applications, the reader may refer to the literature; see [24, 25, 23, 14, 13, 12 ].”
Transverse knots distinguished by knot Floer homology (2007, background): “Although some knot types, including the unknot [10], torus knots [13], and the figure eight knot [13], are known to be Legendrian and transversely simple, it has been known since the work of Chekanov and Eliashberg in the mid 1990’s that not all knots are Legendrian simple.”
Lagrangian concordance of Legendrian knots (2006, methodology): “Using a forthcoming result of Ekholm, Honda and Kálmán [7] we can show that there exists concordances which do not come from Legendrian isotopy.”
Maximal Thurston-Bennequin Number of Two-Bridge Links (2000, background): “imum degree in the framing variable a. With the normalizations of [FT], the Kauﬀman bound states that tb(K) ≤ min-deg aF K(a,x)− 1. The Kauﬀman inequality is not sharp in general; see, e.g., [Fer] or [EH]. Sharpness has been established, however, for some small classes ofknots, including positive kn”
Computable Legendrian invariants (2000, methodology): “wers of t; the DGA is then an algebra over Z graded over Z/(2r(K)), still a lifting of Chekanov’s original DGA over Z/2. As a ﬁnal remark, if K is a stabilization, i.e., contains a zigzag (see, e.g., [6]), then it is easy to see that there is an ai such that ∂ai = 1 or ∂ai = t−1. In this case, ∂(aj ”
On the equivalence of Legendrian and transverse invariants in knot Floer homology (2011, background): “Conversely, any two Legendrian approximations of the same transverse link become Legendrian isotopic after sufficiently many negative Legendrian stabilizations (Etnyre and Honda [14], and Epstein, Fuchs and Meyer [11]).”
Grid Homology for Knots and Links (2015, methodology): “Legendrian simplicity of various knot types (including the unknot [39], torus knots and the figure-eight knot [44, 46]) has been established using cut-andpaste methods in contact topology, relying on convex surface theory [73] and bypasses [90].”
Contact homology and one parameter families of Legendrian knots (2004, background): “The positive Legendrian (p, q) torus knot obtained as a special case is the one with maximal Thurston–Bennequin number p(q−1)− q (see [11] for a classification of Legendrian torus knots).”
Legendrian knots, transverse knots and combinatorial Floer homology (2006, result): “By a result of Epstein, Fuchs and Meyer [4] (generalized by Etnyre and Honda [6]), any two Legendrian approximations to some given transverse knot are equal after some number of negative stabilizations.”
A Legendrian Thurston-Bennequin bound from Khovanov homology (2005, methodology): “It is, however, not sharp for many negative torus knots, for which tb has been computed by Etnyre and Honda [6] using symplectic-topological techniques.”