Cabling and transverse simplicity

Abstract

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]).