Zigzag structure of complexes

Zigzag structure of thin chamber complexes

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 $α_{n}$, 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. Also, we discuss the following problem: in which cases two faces in a thin chamber complex can be connected by a zigzag?

On the lengths of zigzags in thin complexes

We consider zigzags in thin complexes. The main result states that the sum of the lengths of all zigzags in an $n$-complexe is equal to the sum of the lengths of all zigzags in all $(n-1)$-faces of this complex, and this sum also is the twice of the sum of the lengths of all zigzags in all $(n-2)$-faces. For simplicial and cubical $n$-complexes, the sum depends on the rank $n$ and the number of $(n-1)$-faces only. We also describe the sum of the lengths of all generalized zigzags, it depends on the rank and the number of flags. As an application, we find the number of zigzags in Coxeter comple…

Skeletal Geometric Complexes and Their Symmetries

Skeletal polyhedra and polygonal complexes are finite or infinite periodic structures in 3-space with interesting geometric, combinatorial, and algebraic properties. These structures can be viewed as finite or infinite periodic graphs (nets) equipped with a polyhedral superstructure imposed by the faces, allowed to be skew, zigzag, or helical. The article describes skeletal structures with maximal symmetry.

Zigzag Structure of Simple Two-faced Polyhedra

A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbors on opposite edges. A graph without a railroad is called tight. We consider the zigzag and railroad structures of general 3-valent plane graph and, especially, of simple two-faced polyhedra, i.e., 3-valent 3-polytopes with only $a$-gonal and $b$-gonal faces, where $3 \le a < b \le 6$; the main cases are $(a,b)=(3,6)$, $(4,6)$ and $(5,6)$ (the fullerenes). We com…

交替（ zigzag ）数 $A_n$ 的积分和算术结构

Integral and arithmetic structures of alternating (zigzag) numbers $A_n$

本文通过积分和算术结构研究交替数 $A_n$，结合谱展开与斯特林恒等式，推导出其双曲积分表示和同余关系，揭示其分析与组合的双重特性。

Zigzag Stacks and m-Regular Linear Stacks

The contact map of a protein fold is a graph that represents the patterns of contacts in the fold. It is known that the contact map can be decomposed into stacks and queues. RNA secondary structures are special stacks in which the degree of each vertex is at most one and each arc has length at least two. Waterman and Smith derived a formula for the number of RNA secondary structures of length $n$ with exactly $k$ arcs. Höner zu Siederdissen et al. developed a folding algorithm for extended RNA secondary structures in which each vertex has maximum degree two. An equation for the generating func…