The Thurston norm via Normal Surfaces

TL;DR

This paper presents a method to compute the Thurston norm for 3-manifolds using normal surface theory, providing algorithms for calculating the norm, its unit ball, and fibred faces, with bounds on complexity.

Contribution

It introduces a convex polytope model for the Thurston norm and algorithms for computing and analyzing it based on triangulations.

Findings

Algorithm to compute the Thurston norm ball

Exponential bound on vertices of the norm ball

Algorithm to determine fibred faces

Abstract

Given a triangulation of a closed, oriented, irreducible, atoroidal 3-manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by non-negative integer weights, 14 for each 3-simplex, that describe how many copies of each oriented normal disc type there are. The Euler characteristic and homology class are both linear functions of the weights. There is a convex polytope in the space of weights, defined by linear equations given by the combinatorics of the triangulation, whose image under the homology map is the unit ball, B, of the Thurston norm. Applications of this approach include (1) an algorithm to compute B and hence the Thurston norm of any homology class, (2) an explicit exponential bound on the number of vertices of B in terms of the number of simplices in the triangulation, (3) an algorithm to determine the fibred faces of B and hence an algorithm to decide whether a 3-manifold fibres over the circle.