The Thurston norm via normal surfaces

Cooper, Daryl and Tillmann, Stephan (2009) The Thurston norm via normal surfaces. Pacific Journal of Mathematics, 239 1: 1-15. doi:10.2140/pjm.2009.239.1

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Author Cooper, Daryl
Tillmann, Stephan
Title The Thurston norm via normal surfaces
Journal name Pacific Journal of Mathematics   Check publisher's open access policy
ISSN 0030-8730
Publication date 2009-01
Sub-type Article (original research)
DOI 10.2140/pjm.2009.239.1
Open Access Status Not Open Access
Volume 239
Issue 1
Start page 1
End page 15
Total pages 15
Place of publication Berkeley, CA, United States
Publisher University of California, Berkeley
Language eng
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 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.
Keyword 3-Manifolds
Thurston norm
Triangulation
Normal surface
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
ERA 2012 Admin Only
 
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Created: Thu, 01 Dec 2011, 14:31:13 EST by Dr Benjamin Burton on behalf of Mathematics