Geometry & Topology

Bounds for the genus of a normal surface

William Jaco, Jesse Johnson, Jonathan Spreer, and Stephan Tillmann

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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.

Article information

Geom. Topol., Volume 20, Number 3 (2016), 1625-1671.

Received: 24 November 2014
Accepted: 17 July 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57Q15: Triangulating manifolds 57M20: Two-dimensional complexes 57N35: Embeddings and immersions 53A05: Surfaces in Euclidean space

3–manifold normal surface minimal triangulation efficient triangulation realisation problem


Jaco, William; Johnson, Jesse; Spreer, Jonathan; Tillmann, Stephan. Bounds for the genus of a normal surface. Geom. Topol. 20 (2016), no. 3, 1625--1671. doi:10.2140/gt.2016.20.1625.

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