Cabling and transverse simplicity

By John B. Etnyre, K. Honda. 2003

110 citations recorded by the source corpus.

We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type /C, we analyze the Legendrian knots in knot types ob tained from K by cabling, in terms of Legendrian knots in the knot type K. As a corollary of this analysis, we show that the (2,3)-cable of the (2,3)-torus knot is not transversely simple and moreover classify the transverse knots in this knot type. This is the first classification of transverse knots in a non transversely-simple knot type. We also classify Legendrian knots in this knot type and exhibit the first example of a Legendrian knot that does not destabi lize, yet its Thurston-Bennequin invariant is not maximal among Legendrian representatives in its knot type. In this paper we continue the investigation of Legendrian knots in tight contact 3-manifolds using 3-dimensional contact-topological methods. In [EH1], the authors introduced a general framework for analyzing Legendrian knots in tight contact 3-manifolds. There we streamlined the proof of the classification of Legendrian unknots, originally proved by Eliashberg-Eraser in [EF], and gave a complete classification of Legendrian torus knots and figure eight knots. In [EH2], we gave the first structure theorem for Legendrian knots, namely the reduction of the analysis of connected sums of Legendrian knots to that of the prime summands. This yielded a plethora of non-Legendrian-simple knot types. (A topological knot type is Legendrian simple if Legendrian knots in this knot type are determined by their Thurston-Bennequin invariant and rotation number.) Moreover, we exhibited pairs of Legendrian knots in the same topological knot type with the same Thurston-Bennequin and rotation numbers, which required arbitrarily many stabilizations before they became Legendrian isotopic (see [EH2]).

Subfields: Genetics, Geometry and Topology, Neurology

View Paper: https://www.semanticscholar.org/paper/e56ae49ffaee50cb1ae57a38f0f767889dfe7076

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Citations in the corpus

Stabilization in the braid groups II: Transversal simplicity of knots (2003, methodology): “After a draft of this paper was posted and submitted for publication, Etnyre and Honda proved the existence of examples, although they could not make them explicit [9].”
Transverse knots and Khovanov homology (2004, methodology): “However, the existing examples are sparse (those of [ EH3 ] are not even explicit), and the proofs in [BM] and [EH3] require a subtle analysis of braids and contact manifolds.”
Cabling, contact structures and mapping class monoids (2010, background): “Understanding the size of a standard neighborhood of a transverse (or Legendrian) knot has been an important, recurring theme in contact geometry, see [20, 21, 24].”
Legendrian and transverse twist knots (2010, methodology): “Lemma 2.3 (Honda [15]).”
Contact surgeries and the transverse invariant in knot Floer homology (2008, background): “Keywords: Legendrian knots; transverse knots; Heegaard Floer homology AMS 2010 Mathematics subject classification: Primary 57M27 Secondary 57R58”
Grid Homology for Knots and Links (2015, methodology): “While Etnyre and Honda verified transverse non-simplicity for a certain cable of the trefoil knot [47], in this book we will show transverse non-simplicity of several other simpler knot types, including m(10132) shown in Figure 1.”
Heegard Floer invariants of Legendrian knots in contact three-manifolds (2008, unknown): “Transversely non-simple knot types were recently found by Etnyre–Honda and Birman–Menasco [15, 1] (cf.”
Braids: A Survey (2004, background): “ws that no such transversal isotopy exists, a contradiction of our assumption that TX + and TX− are transversally isotopic. k 43 Other examples of a similar nature were discovered by Etnyre and Honda [61] after the proof of Theorem 11 was posted on the arXiv. Their methods are very diﬀerent from the”
Transverse knots distinguished by knot Floer homology (2007, methodology): “three (Birman and Menasco [3, 2]) and the (2, 3) cable of the (2, 3) torus knot (Etnyre and Honda [14]).”
Fibered Transverse Knots and the Bennequin Bound (2008, background): “For example, the (2,3)-cable of the (2,3)-torus knot is a fibered knot that supports the tight contact structure on S3, but in [ 11 ] it was”
Contact homology and one parameter families of Legendrian knots (2004, result): “Chekanov [6] introduced a much more complicated, Floer-theoretical invariant which was able to distinguish between Legendrian knots of the same tb, r, and classical type (since then, Etnyre and Honda [11, 12, 13] have been able to prove similar results with diﬀerent methods). The form in which we’ll”