几乎法向曲面的四边形-八边形坐标
Quadrilateral-octagon coordinates for almost normal surfaces
摘要：
法向曲面和几乎法向曲面是算法三维流形拓扑的基本工具，但要使用它们需要高维向量空间中指数缓慢的枚举算法。Tollefson的四边形坐标通过将这个向量空间的维度从7 n减少到3 n（其中n是底层三角剖分的复杂度），大大减轻了正常曲面的这个问题。在这里，我们开发了一个类似的理论，八边形几乎正常的表面，使用四边形和八边形坐标，以减少这方面的10 n到6 n。作为一个应用程序，我们表明，四边形，八边形坐标可以专门用于流线型3球识别算法的雅科，鲁宾斯坦和汤普森，减少实验运行时间的数千倍。我们还介绍了联合坐标，系统只有3 n个维度的八角形几乎正常的表面，具有吸引人的几何性质。
英文摘要：
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.
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