Cabling and Transverse Simplicity

We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type ${\cal K}$, we analyze the Legendrian knots in knot types obtained from ${\cal K}$ by cabling, in terms of Legendrian knots in the knot type ${\cal K}$. 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.