Voltage operations on maniplexes

TL;DR

This paper introduces voltage operations on maniplexes, a generalization of classical geometric operations, enabling the construction and analysis of maniplexes with specific symmetry properties using graph theory techniques.

Contribution

It defines voltage operations on maniplexes, linking classical geometric operations to graph theory, and characterizes when an operation can be viewed as a voltage operation.

Findings

Voltage operations can be applied to symmetry type graphs.

They help build maniplexes with prescribed symmetry types.

A complete characterization of voltage operations is provided.

Abstract

Classical geometric and topological operations on polyhedra, maps and polytopes often give rise to structures with the same symmetry group as the original one, but with more flags. In this paper we introduce the notion of voltage operations on maniplexes, as a way to generalize such operations. Our technique provides a way to study classical operations in a graph theory setting. In fact, voltage operations can be applied to symmetry type graphs, and more generally to n-valent properly n-edge colored graphs. We focus on studying the interactions between voltage operations and the symmetries of the operated object, and show that they can be potentially used to build maniplexes with prescribed symmetry type graphs. Moreover, a complete characterization of when an operation can be seen as a voltage operation is given.