[论文解读] Maximizing the number of edges in three-dimensional colored triangulations whose building blocks are balls

Valentin Bonzom|arXiv (Cornell University)|Feb 18, 2018

Advanced Combinatorial Mathematics参考文献 57被引用 11

一句话总结

本文解决了在所有但至多一个构建块均为3-球的3D彩色三角剖分中最大化边数的问题。证明了此类边最大化的三角剖分同胚于3-球，并与对偶1-骨架中的树一一对应，边数通过构建块上的独立求和计算得出。关键结果是通过对偶图中的2-边割对类球形块进行组合表征。

ABSTRACT

Colored triangulations offer a generalization of combinatorial maps to higher dimensions. Just like maps are gluings of polygons, colored triangulations are built as gluings of special, higher-dimensional building blocks, such as octahedra, which we call colored building blocks and known in the dual as bubbles. A colored building block is determined by its boundary triangulation, which in the case of polygons is simply characterized by its length. In three dimensions, colored building blocks are labeled by some 2D triangulations and those homeomorphic to the 3-ball are labeled by the subset of planar ones. Similarly to Euler's formula in 2D which provides an upper bound to the number of vertices at fixed number of polygons with given lengths, we look in three dimensions for an upper bound on the number of edges at fixed number of given colored building blocks. In this article we solve this problem when all colored building blocks, except possibly one, are homeomorphic to the 3-ball. To do this, we find a characterization of the way a colored building block homeomorphic to the ball has to be glued to other blocks of arbitrary topology in a colored triangulation which maximizes the number of edges. This local characterization can be extended to the whole triangulation as long as there is at most one colored building block which is not a 3-ball. The triangulations obtained this way are in bijection with trees. The number of edges is given as an independent sum over the building blocks of such a triangulation. In the case of all colored building blocks being homeomorphic to the 3-ball, we show that these triangulations are homeomorphic to the 3-sphere. Those results were only known for the octahedron and for melonic building blocks before. This article is self-contained and can be used as an introduction to colored triangulations and their bubbles from a purely combinatorial point of view.

研究动机与目标

• 确定由彩色构建块组成的3维彩色三角剖分中可能实现的最大边数，其中所有但可能一个块同胚于3-球。
• 表征此类边最大化的三角剖分中3-球构建块的粘合结构。
• 在边最大化的三角剖分与对偶1-骨架中的树之间建立双射。
• 证明当所有构建块均为3-球时，所得三角剖分为同胚于3-球。
• 将先前仅针对正八面体和梅洛尼块的结果推广至任意平面三角剖分为构建块。

提出的方法

• 本文使用彩色三角剖分的对偶表示，即边着色图，其中3-泡对应于顶点，2-边割对应于类球形块的拓扑切除。
• 采用基于边翻转和2-二面体收缩的递归论证，以在保持拓扑类型和最大化边数的同时简化图。
• 该方法依赖于对3-泡（对偶于顶点）应用欧拉公式，推导出双色圈和分量亏格的约束。
• 证明边最大化的图必须具有所有3-泡为平面图（亏格为零），利用边翻转在特定条件下保持双色圈数量的事实。
• 关键技术工具是通过2-边割对拓扑操作的表征，表明当且仅当通过此类割粘合时，类球形块可从三角剖分中切除。
• 该构造被扩展以证明当所有块均为球体时，整个三角剖分为球面，通过从初始图到已知球面图的一系列拓扑操作。

实验结果

研究问题

• RQ1 当所有但一个构建块同胚于3-球时，3D彩色三角剖分中可实现的最大边数是多少？
• RQ2 3-球构建块必须如何粘合到其他块上，以使三角剖分的边数最大化？
• RQ3 边最大化的性质能否通过对偶1-骨架和2-边割进行组合表征？
• RQ4 在何种条件下整个三角剖分为同胚于3-球？
• RQ5 是否存在边最大化的三角剖分与对偶图中组合树之间的双射？

主要发现

• 边最大化的三角剖分中的边数由构建块上的独立求和给出，其贡献由其内部结构和边界三角剖分决定。
• 同胚于3-球的构建块必须以可通过对偶1-骨架中一系列2-边割切除的方式粘合。
• 当所有构建块均为3-球时，所得三角剖分为同胚于3-球。
• 边最大化的三角剖分与树之间存在双射，其中每个节点对应一个构建块，每条边对应一个2-边割连接。
• 此类三角剖分中的3-泡（对偶于顶点）均为平面图，即其亏格为零，这是边最大化的必要条件。
• 该构造将先前针对正八面体和梅洛尼块的结果推广至任意平面三角剖分为构建块。