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Bounds for the genus of a normal surface

William Jaco, Jesse Johnson, Jonathan Spreer and Stephan Tillmann

Geometry & Topology 20 (2016) 1625–1671

This paper gives sharp linear bounds on the genus of a normal surface in a triangulated compact orientable 3–manifold in terms of the quadrilaterals in its cell decomposition — different bounds arise from varying hypotheses on the surface or triangulation. Two applications of these bounds are given. First, the minimal triangulations of the product of a closed surface and the closed interval are determined. Second, an alternative approach to the realisation problem using normal surface theory is shown to be less powerful than its dual method using subcomplexes of polytopes.

Dedicated to Hyam Rubinstein on the occasion of his 65\textitth birthday

3–manifold, normal surface, minimal triangulation, efficient triangulation, realisation problem
Mathematical Subject Classification 2010
Primary: 57N10
Secondary: 57Q15, 57M20, 57N35, 53A05
Received: 24 November 2014
Accepted: 17 July 2015
Published: 4 July 2016
Proposed: Cameron Gordon
Seconded: David Gabai, Ronald Stern
William Jaco
Department of Mathematics
Oklahoma State University
Stillwater, OK 74078-1058
Jesse Johnson
Mathematics Department
Oklahoma State University
Stillwater, OK 74078-1058
Jonathan Spreer
School of Mathematics and Physics
The University of Queensland
Brisbane, QLD 4072
Stephan Tillmann
School of Mathematics and Statistics
The University of Sydney
Sydney, NSW 2006