About the existence of a universal transverse knot

Abstract

Since 2002 thanks to Giroux it is known that any 3-dimensional contact manifold $(M,\xi )$ can be obtained as 3-fold simple covering $f:M\to S^3$ branched along a link transverse to the standard contact structure of $S^3$ in such a way that induced contact structure by $f$ on $M$ is isotopic to $\xi$.

Recently, Cassals and Etnyre (2018) proved that it is possible to fix the link if we remove the conditions of simplicity and 3-folds. They also found a 5 component link with this property and which they called transverse universal. They also asked if there exists a universal transverse knot (one component link).

On this session, we will work on the above question.