Platonic Solids and High Genus Covers of Lattice Surfaces
Platonic Solids and High Genus Covers of Lattice Surfaces
复制标题
柏拉图实体和格子曲面的高格覆盖
作者：
J. Athreya;D. Aulicino;W. Hooper
摘要我们研究了通过考虑柏拉图固体表面的展开而得到的平移面。我们证明了它们都是格子曲面，并计算了伴随的Teichmüler曲线的拓扑。利用一个计算原始格子曲面平移覆盖的Teichmüler曲线的通用算法，我们证明了展开的十二面体的Teichmüler曲线具有亏格131，具有19个锥奇点和362个尖点。我们提供了理论和严格的计算机辅助证明，证明了在与四面体、八面体、立方体和二十面体相关的曲面上没有闭鞍点。我们证明了十二面体上的闭鞍环恰好存在31个等价类，其中等价定义为平移覆盖的仿射自同构。这里建立的技巧更一般地适用于柏拉图曲面，甚至更一般地适用于原始格子曲面及其欧几里得锥面和台球台商的平移覆盖。
Abstract We study the translation surfaces obtained by considering the unfoldings of the surfaces of Platonic solids. We show that they are all lattice surfaces and we compute the topology of the associated Teichmüller curves. Using an algorithm that can be used generally to compute Teichmüller curves of translation covers of primitive lattice surfaces, we show that the Teichmüller curve of the unfolded dodecahedron has genus 131 with 19 cone singularities and 362 cusps. We provide both theoretical and rigorous computer-assisted proofs that there are no closed saddle connections on the surfaces associated to the tetrahedron, octahedron, cube, and icosahedron. We show that there are exactly 31 equivalence classes of closed saddle connections on the dodecahedron, where equivalence is defined up to affine automorphisms of the translation cover. Techniques established here apply more generally to Platonic surfaces and even more generally to translation covers of primitive lattice surfaces and their Euclidean cone surface and billiard table quotients.