Existence of a transverse universal knot

Abstract

We prove that there is a knot $k$ transverse to $\xi_{std}$, the tight contact structure of $S^3$, such that every contact 3-manifold $(M,\xi)$ can be obtained as a contact covering branched along $k$. By contact covering we mean a map $\phi:M \to S^3$ branched along $k$ such that $\xi$ is contact isotopic to the lifting of $\xi_{std}$ under $\phi$.