# 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
United States

Ko Honda

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