Core curves of triangulated solid tori

TL;DR

The paper proves that in any triangulation of a solid torus, a simple pre-core curve exists with limited face intersections, impacting Riemannian metrics and knot theory applications.

Contribution

It establishes the existence of a pre-core curve with bounded face intersections in any triangulation of a solid torus, a new geometric topological result.

Findings

Existence of a pre-core curve with at most 10 intersections per face

Restrictions on Riemannian metrics on solid tori

Applications to knot theory

Abstract

We show that in any triangulation of a solid torus, there is a pre-core curve that lies in the 2-skeleton and that intersects the interior of each face in at most 10 straight arcs. By definition, a pre-core curve is a simple closed curve that becomes a core curve when a collar is attached to the boundary of the solid torus. This theorem imposes restrictions on the possible Riemannian metrics on a solid torus. It also has applications in knot theory.