The existence of a universal transverse knot

Abstract

We prove that there is a knot $K$ transverse to ${\xi }_{\text{std}}$, 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 ${\xi }_{\text{std}}$ under $\phi$.