Quadrilateral-octagon coordinates for almost normal surfaces

Burton, Benjamin A. (2010) Quadrilateral-octagon coordinates for almost normal surfaces. Experimental Mathematics, 19 3: 285-315. doi:10.1080/10586458.2010.10390625

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Author Burton, Benjamin A.
Title Quadrilateral-octagon coordinates for almost normal surfaces
Journal name Experimental Mathematics   Check publisher's open access policy
ISSN 1058-6458
1944-950X
Publication date 2010
Year available 2011
Sub-type Article (original research)
DOI 10.1080/10586458.2010.10390625
Volume 19
Issue 3
Start page 285
End page 315
Total pages 31
Place of publication Natick, MA, U.S.A.
Publisher A K Peters
Collection year 2011
Language eng
Formatted abstract
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.
Keyword Normal surfaces
Almost normal surfaces
Quadrilateral-octagon coordinates
Joint coordinates
Q-theory
3-sphere recognition
Q-Index Code C1
Q-Index Status Confirmed Code
Institutional Status UQ
Additional Notes Available online: 11 Feb 2011

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
Non HERDC
 
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Created: Sun, 05 Dec 2010, 00:12:43 EST