Cabling and transverse simplicity (Q2494226)

The authors study cables of oriented Legendrian and transverse knots in the standard tight contact \((S^3, \xi_{\text{std}}).\) For a topological knot type \(K,\) put \(\overline{tb}(K)\) to be the maximal value of the Thurston-Bennequin invariant of a Legendrian knot realizing \(K\) and put \(K_{(p,q)}\) to be the \((p,q)\)-cable of \(K.\) A knot type \(K\) is Legendrian simple if Legendrian knots realizing \(K\) are determined by their self-linking and rotation numbers; and it is transversely simple if transverse knots realizing \(K\) are determined by their self-linking number. An embedding \(\phi:S^1\times D^2\to S^3\) is said to represent \(K\) if the core curve of \(\phi\) represents \(K.\) The slope of homotopically nontrivial curves on \(\partial(S^1\times D^2)=\mathbb R^2/\mathbb Z^2\) is measured via the convention that the meridian has slope \(0\) and the longitude has slope \(\infty.\) Define the width \(w(K)=\sup \frac{1}{\text{slope}(\Gamma_{\partial(S^1\times D^2)})}.\) Here \(\Gamma_{\partial(S^1\times D^2)}\) is the dividing set and the supremum is taken over all embeddings \(S^1\times D^2\to S^3\) representing \(K\) with convex \(\partial(S^1\times D^2)\). We say that \(K\) has the uniform thickness property (UTP) if the following two conditions 1 and 2 hold. 1: \(\overline{tb}(K)=w(K).\) (It is always true that \(\overline{tb}(K)\leq w(K)\leq \overline tb(K)+1\).) 2: every embedded \(S^1\times D^2\) representing \(K\) can be thickened to a standard neighborhood of a maximal \(tb\) Legendrian knot. (A standard neighborhood \(N(L)\) of a Legendrian knot \(L\) is an embedded solid torus with convex boundary and core \(L\) such that \(\#\Gamma_{\partial N(L)}=2\) and \(tb (L)=\frac{1}{\text{slope} (\Gamma _{\partial N(L)})}.\)) The authors prove that if \(K\) is Legendrian simple and satisfies UTP, then \(K_{(p,q)}\) is Legendrian simple. They prove that if \(K_1, K_2\) satisfy UTP, then so does \(K_1\# K_2\); and that if \(K\) satsifies UTP, then so does \(K_{(p,q)}\) for \(\frac{p}{q}<w(K).\) They also show that negative torus knots satisfy UTP and that some positive torus knots do not satisfy UTP. Using these techniques the authors classify all the transverse knots realizing a \((2,3)\)-cable of a \((2,3)\)-torus knot and show that the cable is not transversally simple. This is the first classification of transverse knots in a non-transversely simple knot type. Previously knots that are not transversely simple were found by \textit{J. S. Birman} and \textit{W. W. Menasco} using different techniques. Their results were published in [Geom. Topol. 10, 1425--1452 (2006; Zbl 1130.57005)].