The existence of a universal transverse 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 $\varphi: M \to S^3$ branched along $K$ such that $\xi$ is contact isotopic to the lifting of $\xi_{std}$ under $\varphi$.