The 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 $\phi$.

36 pages, 22 figures To appear on Algebraic and Geometric Topology