The Thurston norm via Normal Surfaces
Abstract
Given a triangulation of a closed, oriented, irreducible, atoroidal 3manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by nonnegative integer weights, 14 for each 3simplex, 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 3manifold fibres over the circle.
 Publication:

arXiv eprints
 Pub Date:
 June 2007
 DOI:
 10.48550/arXiv.0706.0673
 arXiv:
 arXiv:0706.0673
 Bibcode:
 2007arXiv0706.0673C
 Keywords:

 Mathematics  Geometric Topology;
 57M25;
 57N10
 EPrint:
 13 pages, 4 figures