Quadrilateral-octagon coordinates for almost normal surfaces - Mathematics > Geometric TopologyReport as inadecuate




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Abstract: Normal and almost normal surfaces are essential tools for algorithmic3-manifold topology, but to use them requires exponentially slow enumerationalgorithms in a high-dimensional vector space. The quadrilateral coordinates ofTollefson alleviate this problem considerably for normal surfaces, by reducingthe dimension of this vector space from 7n to 3n where n is the complexity ofthe underlying triangulation. Here we develop an analogous theory foroctagonal almost normal surfaces, using quadrilateral and octagon coordinatesto reduce this dimension from 10n to 6n. As an application, we show thatquadrilateral-octagon coordinates can be used exclusively in the streamlined3-sphere recognition algorithm of Jaco, Rubinstein and Thompson, reducingexperimental running times by factors of thousands. We also introduce jointcoordinates, a system with only 3n dimensions for octagonal almost normalsurfaces that has appealing geometric properties.



Author: Benjamin A. Burton

Source: https://arxiv.org/







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