Zigzag Structure of Thin Chamber Complexes

Abstract

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 zigzags in the simplex α n , the cross-polytope β 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. We also discuss in which cases two faces in a thin chamber complex can be connected by a zigzag.

On two types of Z-monodromy in triangulations of surfaces

“…is the face shadow of the reversed zigzag. 4 The next two edges from Ω(F ) ∪ Ω(F ′ ) contained in the zigzag Z are M F ′ (e 1 ) = e ′ 2 and D F ′ (e ′ 2 ) = e ′ 3 .…”

“…Zigzags are known also as Petrie paths [2] or closed left-right paths [5,10]. Similar objects in simplicial complexes and abstract polytopes are investigated in [4,11]. An embedded graph is called z-knotted if it contains a single zigzag.…”

Geometric Structure of Chemistry-Relevant Graphs

“…In this chapter, based mainly on [DeDu04], we focus on generalization of zigzags for higher dimension. Inspired by Coxeter's notion of Petrie polygon for d-polytopes (see [Cox73]), we generalize the notion of zigzag circuits on complexes and compute the zigzag structure for several interesting families of d-polytopes, including semiregular, regular-faced, Wythoff Archimedean ones, Conway's 4-polytopes, half-cubes, and folded cubes.…”

Connected sums of z-knotted triangulations

“…The notion of Petrie polygon for polytopes was introduced in Coxeter's book [1] (see also [4,6,11] and [5,Chapter 8] for some generalizations). For graphs embedded in 2-dimensional surfaces the same objects appear as zigzags in [2,3,5], geodesics in [7] and left-right paths in [10].…”

“…For graphs embedded in 2-dimensional surfaces the same objects appear as zigzags in [2,3,5], geodesics in [7] and left-right paths in [10]. Following [2,3,4,5] we call them zigzags.…”