Meridional Almost Normal Surfaces in Knot Complements

TL;DR

This paper proves that in certain 3-manifold knot complements, weakly incompressible bridge surfaces with a bounded number of bridges can be isotoped to almost normal surfaces within a specially chosen triangulation.

Contribution

It establishes the existence of a triangulation where all weakly incompressible bridge surfaces up to a given bridge number are isotopic to almost normal surfaces.

Findings

Existence of special triangulations for knot complements.

Weakly incompressible bridge surfaces are isotopic to almost normal surfaces.

Results apply to knots with irreducible complements.

Abstract

Suppose $K$ is a knot in a closed 3-manifold $M$ such that $\bar{M-N(K)}$ is irreducible. We show that for any positive integer $b$ there exists a triangulation of $\bar{M-N(K)}$ such that any weakly incompressible bridge surface for $K$ of $b$ bridges or fewer is isotopic to an almost normal bridge surface.