Algebraic & Geometric Topology

Converting between quadrilateral and standard solution sets in normal surface theory

Benjamin A Burton

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The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3–manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson’s Q–theory speeds up this operation by using a much smaller space (quadrilateral coordinates), at the cost of a reduced solution set that might not always be sufficient for our needs. In this paper we present algorithms for converting between solution sets in quadrilateral and standard coordinates. As a consequence we obtain a new algorithm for enumerating all standard vertex normal surfaces, yielding both the speed of quadrilateral coordinates and the wider applicability of standard coordinates. Experimentation with the software package Regina shows this new algorithm to be extremely fast in practice, improving speed for large cases by factors from thousands up to millions.

Article information

Algebr. Geom. Topol., Volume 9, Number 4 (2009), 2121-2174.

Received: 23 February 2009
Accepted: 1 September 2009
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 52B55: Computational aspects related to convexity {For computational geometry and algorithms, see 68Q25, 68U05; for numerical algorithms, see 65Yxx} [See also 68Uxx]
Secondary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 57N35: Embeddings and immersions

normal surfaces Q-theory vertex enumeration conversion algorithm double description method


Burton, Benjamin A. Converting between quadrilateral and standard solution sets in normal surface theory. Algebr. Geom. Topol. 9 (2009), no. 4, 2121--2174. doi:10.2140/agt.2009.9.2121.

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