Universal knots

Detecting what makes a knot unversal

Description

We say that a knot is universal if any compact, orientable 3-manifold can be obtained as a covering of $\mathbb{S}^3$ branched along that knot. Our goal is to understand the underlying properties of a knot that make it universal. While many knots have been proven to be universal, we currently lack a definitive tool to determine whether a given knot is universal. It is conjectured that every hyperbolic knot is universal.

The universality of two-bridge knots was established by Hilden, Lozano, and Montesinos in 1985. They proved that every hyperbolic two-bridge knot is universal. Specifically, the figure-eight knot is universal, whereas the trefoil knot is not. It is straightforward to prove that every torus knot is not universal. The reason lies in the fact that torus knots can be viewed as fibers in a Seifert fibration of $\mathbb{S}^3$. Such fibrations lift naturally to any branched covering, implying that only Seifert fibered manifolds are obtained from torus knots.

A natural question arises: does hyperbolicity imply universality? Using dihedral branched coverings, we were able to verify this conjecture for all Montesinos knots with up to nine crossings, except for two specific knots, the knots $9_{35}$ and $9_{48}$. The former was later proven to be universal by Nuñez and Jordán in their paper titled “Classical Drawings of Branched Coverings” The question of the universality of $9_{48}$ remains open.

Universality in Contact Manifolds

A similar problem arises in the study of contact manifolds. By considering the branched covering of a transverse knot in a contact manifold, we can naturally lift the contact structure to the covering space. Thus, we define a transverse knot in standard contact $\mathbb{S}^3$ to be universal if any contact-orientable 3-manifold can be obtained as a branched covering of $\mathbb{S}^3$ along that knot. These knots are sometimes referred to as contact universal or transverse universal.

It has been proven that universal transverse knots exist, but the figure-eight knot is not one of them. This indicates that universality in the context of contact manifolds is more restrictive. We are still searching for the simplest knot type with a universal transverse representative.

Recently, Zapata-Rendón published a result showing that many twist knots are not universal. This leaves the mirror image of the $5_2$ knot as the smallest remaining candidate for a transverse universal knot. Unlike the figure-eight knot, this example has tight contact manifolds as branched coverings.