On the lengths of zigzags in thin complexes

TL;DR

This paper investigates the properties of zigzags in thin complexes, revealing relationships between their lengths across different faces and applying these findings to Coxeter complexes.

Contribution

It establishes a fundamental relation between zigzag lengths in thin complexes and provides formulas for simplicial, cubical, and Coxeter complexes.

Findings

Sum of zigzag lengths in an n-complex equals the sum in all (n-1)-faces.

Sum of zigzag lengths in an n-complex is twice the sum in all (n-2)-faces.

Explicit formulas for zigzag sums in Coxeter complexes.

Abstract

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 complexes.