Zigzag structure of thin chamber complexes

TL;DR

This paper explores zigzags in thin chamber complexes, describing their structure in Coxeter complexes and relating their lengths to Coxeter numbers across various polytopes, while also addressing face connectivity via zigzags.

Contribution

It provides a comprehensive description of zigzags in Coxeter complexes and establishes their length correspondence with Coxeter numbers for key polytopes.

Findings

All zigzags in Coxeter complexes are described.

Lengths of generalized zigzags match Coxeter numbers for specific polytopes.

Discussion on face connectivity via zigzags in thin chamber complexes.

Abstract

Zigzags and generalized zigzags in thin chamber complexes are investigated, in particular, all zigzags in the Coxeter complexes are described. Using this description, we show that the lengths of all generalized zigzags in the simplex $\alpha_{n}$, the cross-polytope $\beta_{n}$, the $24$-cell, the icosahedron and the $600$-cell are equal to the Coxeter numbers of $A_{n}$, $B_{n}=C_{n}$, $F_{4}$ and $H_{i}$, $i=3,4$, respectively. Also, we discuss the following problem: in which cases two faces in a thin chamber complex can be connected by a zigzag?