Meridional almost normal surfaces in knot complements

Abstract

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