Degree-regular triangulations of the double-torus

Degree-regular triangulations of torus and Klein bottle

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In 1999, Lutz has classified all the weakly regular triangulations on at most 15 vertices. In 2001, Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degree-regular triangul…

紧李群环面的等变三角剖分以及到非晶体学Coxeter群的双曲延拓

Equivariant triangulations of tori of compact Lie groups and hyperbolic extension to non-crystallographic Coxeter groups

该研究针对紧李群极大环面的Weyl群作用，利用仿射Weyl群组合构造等变三角剖分与相关dg-环，还将结果推广到非晶体学Coxeter群，通过紧双曲延拓构造类环面流形并研究其等变结构与同调表示。

Optimal Triangulation of Regular Simplicial Sets

The Barratt nerve, denoted $B$, is the endofunctor that takes a simplicial set to the nerve of the poset of its non-degenerate simplices. The ordered simplicial complex $BSd\, X$, namely the Barratt nerve of the Kan subdivision $Sd\, X$, is a triangulation of the original simplicial set $X$ in the sense that there is a natural map $BSd\, X\to X$ whose geometric realization is homotopic to some homeomorphism. This is a refinement to the result that any simplicial set can be triangulated. A simplicial set is said to be regular if each of its non-degenerate simplices is embedded along its $n$-th …

Generalized Tonnetz and discrete Abel-Jacobi map

Motivated by classical Euler's $Tonnetz$, we introduce and study the combinatorics and topology of more general simplicial complexes $Tonn^{n,k}(L)$ of "Tonnetz type". Out main result is that for a sufficiently generic choice of parameters the generalized tonnetz $Tonn^{n,k}(L)$ is a triangulation of a $(k-1)$-dimensional torus $T^{k-1}$. In the proof we construct and use the properties of a "discrete Abel-Jacobi map", which takes values in the torus $T^{k-1} \cong \mathbb{R}^{k-1}/Λ$ where $Λ\cong \mathbb{A}^\ast_{k-1}$ is the permutohedral lattice.