This paper is an introductory and informal exposition on the topology of polygonal meshes. We begin with a broad overview of topological notions and discuss how homeomorphisms, homotopy, and homology can be used to characterise topology. We move on to define polygonal meshes and make a distinction between intrinsic topology and extrinsic topology which depends on the space in which the mesh is immersed. A distinction is also made between quantitative topological properties and qualitative properties. Next, we outline proofs of the Euler and the Euler-Poincaré formulas. The Betti numbers are then defined in terms of the Euler-Poincaré formula and other mesh statistics rather than as cardinalities of the homology groups which allows us to avoid abstract algebra. Finally, we discuss how it is possible to cut a polygonal mesh such that it becomes a topological disc.

翻译：本文是关于多边形网格拓扑结构的入门性与非正式阐述。我们首先对拓扑概念进行广泛概述，并讨论如何利用同胚、同伦与同调来刻画拓扑特性。随后定义多边形网格，区分依赖于网格浸入空间的外在拓扑与内在拓扑，同时辨析定量拓扑性质与定性拓扑性质。接着，我们概述欧拉公式与欧拉-庞加莱公式的证明过程。继而通过欧拉-庞加莱公式及其他网格统计量定义贝蒂数，而非采用同调群基数定义方式，从而避免抽象代数概念的引入。最后，我们探讨如何通过切割将多边形网格转化为拓扑圆盘。