Veering triangulations and the Thurston norm: homology to isotopy

TL;DR

This paper demonstrates how veering triangulations can determine a face of the Thurston norm ball in a closed 3-manifold and compute the Thurston norm, also classifying taut surfaces up to isotopy.

Contribution

It establishes a direct link between veering triangulations and the Thurston norm, including nonlayered and nonfibered cases, and shows how they classify taut surfaces.

Findings

Veering triangulation specifies a face of the Thurston norm ball.

Veering triangulation computes the Thurston norm in the cone over that face.

Veering triangulation classifies taut surfaces up to isotopy.

Abstract

We show that a veering triangulation $\tau$ specifies a face $\sigma$ of the Thurston norm ball of a closed three-manifold, and computes the Thurston norm in the cone over $\sigma$. Further, we show that $\tau$ collates exactly the taut surfaces representing classes in the cone over $\sigma$ up to isotopy. The analysis includes nonlayered veering triangulations and nonfibered faces.