Non-parallel essential surfaces in knot complements

TL;DR

This paper establishes a relationship between the thin position of knots or links and the existence of multiple essential surfaces in their complements, with implications for triangulation complexity.

Contribution

It introduces a method to find multiple disjoint essential surfaces based on the thin position of knots or links, linking geometric and topological properties.

Findings

Existence of n disjoint essential surfaces for knots with n thin levels.

At least n/3 tetrahedra in any triangulation of the knot complement.

Connection between thin position and triangulation complexity.

Abstract

We show that if a knot or link has n thin levels when put in thin position then its exterior contains a collection of n disjoint, non-parallel, planar, meridional, essential surfaces. A corollary is that there are at least n/3 tetrahedra in any triangulation of the complement of such a knot.