Experimental Mathematics

Quadrilateral–Octagon Coordinates for Almost Normal Surfaces

Benjamin A. Burton

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Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces by reducing the dimension of this vector space from $7n$ to $3n$ (where $n$ is the complexity of the underlying triangulation). Here we develop an analogous theory for octagonal almost normal surfaces, using quadrilateral and octagon coordinates to reduce this dimension from $10n$ to $6n$. As an application, we show that quadrilateral– octagon coordinates can be used exclusively in the streamlined 3-sphere recognition algorithm of Jaco, Rubinstein, and Thompson, reducing experimental running times by factors of thousands. We also introduce joint coordinates, a system with only $3n$ dimensions for octagonal almost normal surfaces that has appealing geometric properties.

Article information

Experiment. Math., Volume 19, Issue 3 (2010), 285-315.

First available in Project Euclid: 4 October 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Normal surfaces almost normal surfaces quadrilateral-octagon coordinates joint coordinates Q-theory 3-sphere recognition


Burton, Benjamin A. Quadrilateral–Octagon Coordinates for Almost Normal Surfaces. Experiment. Math. 19 (2010), no. 3, 285--315. https://projecteuclid.org/euclid.em/1317758093

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