Description

We say that a knot $k \subset \mathbb{S}^3$ is universal if you can obtain any compact orientable 3-manifold as a covering of $\mathbb{S}^3$ branched along $k$. We try to understand the underlying property of a knot that makes it universal. Many knots have been proven universal, but we lack a tool to decide whether a knot is. It is conjectured that every hyperbolic knot is universal.

The universality of two-bridge knots was determined by Hilden, Lozano, and Montesinos in 1985. They proved that every hyperbolic 2-bride knot is universal. In particular, the figure-eight knot is universal, and the trefoil knot is not. It can easily be proven that every torus knot is not universal, and the reason lies in the fact that they can be seen as a fiber of a Seifert fibration of $\mathbb{S}^3$, such fibration can esily be lifted to every branch covering, implying that only Seifert Fiber Manifolds are obtained from those knots.

A natural question to ask is if hyperbolicity implies universality. Using dihedral branch covering, we were able to verify this conjecture for all Montesinos knots up to 9 crossings, except for the knots $9_{35}$ and $9_{48}$, the former it surprisingly symmetrical (pretzel $p(3.3.3)$ ), but whenever we found $\mathbb{S}^3$ as a branch covering, the preimage of $9_{35}$ contain nothing but torus components. There is a good chance that $9_{35}$ is not universal, and maybe it has to do with this $\mathbb{Z}_3$ symmetry action.

Universality on contact manifolds

A similar problem occurs in the world of contact manifolds. By taking the branch covering of a transverse knot on a contact manifold, we can naturally lift the contact structure to the covering space. So, we can define a transverse knot $k$ in the 3-sphere to be universal if we can obtain any contact-orientable 3-manifold as a covering of $\mathbb{S}^3$ branched along $k$. Sometimes we refer to these knots as contact universal or transverse universal.

It has been proved that a universal transverse knot exists and that the figure-eight knot is not one of them. Meaning that universality in this context is more strict.