Cabling and transverse simplicity

Abstract

We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type $\mathcal{K}$, we analyze the Legendrian knots in knot types obtained from $\mathcal{K}$ by cabling, in terms of Legendrian knots in the knot type $\mathcal{K}$. 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 destabilize, yet its Thurston-Bennequin invariant is not maximal among Legendrian representatives in its knot type.

Authors

John B. Etnyre

Department of Mathematics
University of Pennsylvania
Philadelphia, PA 19104
United States

Ko Honda

Department of Mathematics
University of Southern California
Los Angeles, CA 90089
United States