The Thurston norm via spun-normal immersions

TL;DR

This paper introduces a new approach using spun-normal immersions to compute the Thurston norm in cusped hyperbolic 3-manifolds, providing tools for understanding surface complexity and entropy bounds.

Contribution

It develops a theory of spun-normal immersed surfaces, including algorithms for Thurston norm computation and entropy bounds for pseudo-Anosov maps.

Findings

Algorithm for Thurston norm unit ball computation

Linear bounds on minimal entropy of pseudo-Anosov maps

New theoretical framework for immersed surfaces in 3-manifolds

Abstract

A theory of transversely oriented spun-normal immersed surfaces in ideally triangulated 3--manifolds is developed in this paper, including linear functionals determining the boundary curves, Euler characteristic and homology class of these immersions. This is used to develop and implement an algorithm to compute the unit ball of the Thurston norm for cusped hyperbolic 3--manifolds of finite volume. As an application of independent interest, we give an upper bound on the minimal entropy of pseudo-Anosov maps of surfaces with number of cusps bounded linearly in genus.