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 nonnegative 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.
Keywords
3-manifold, Thurston norm, triangulation, normal surface
