Ideal Triangulations and Once-Punctured Surface Bundles

TL;DR

This paper proves the existence of ideal triangulations that normalize fibers in certain 3-manifolds and provides an algorithm for their construction when the manifold has a single boundary component.

Contribution

It establishes the existence of fiber-normalizing ideal triangulations in atoroidal, acylindrical 3-manifolds and offers a constructive algorithm for manifolds with one boundary component.

Findings

Proves existence of fiber-normalizing ideal triangulations.

Provides an explicit algorithm for constructing such triangulations.

Extends Walsh's result to include the existence of these triangulations.

Abstract

A well-known result of Walsh states that if $\mathcal T^*$ is an ideal triangulation of an atoroidal, acylindrical, irreducible, compact 3-manifold with torus boundary components, then every properly embedded, two-sided, incompressible surface $S$ is isotopic to a spun-normal surface unless $S$ is isotopic to a fiber or virtual fiber. Previously it was unknown if for such a 3-manifold an ideal triangulation in which a fiber spun-normalizes exists. We give a proof of existence and give an algorithm to construct the ideal triangulation provided the 3-manifold has a single boundary component.