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2009 Converting between quadrilateral and standard solution sets in normal surface theory
Benjamin A Burton
Algebr. Geom. Topol. 9(4): 2121-2174 (2009). DOI: 10.2140/agt.2009.9.2121


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.


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Benjamin A Burton. "Converting between quadrilateral and standard solution sets in normal surface theory." Algebr. Geom. Topol. 9 (4) 2121 - 2174, 2009.


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

zbMATH: 1198.57013
MathSciNet: MR2551665
Digital Object Identifier: 10.2140/agt.2009.9.2121

Primary: 52B55
Secondary: 57N10, 57N35

Rights: Copyright © 2009 Mathematical Sciences Publishers


Vol.9 • No. 4 • 2009
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