Characterizing and Detecting Toric Loops in n-Dimensional Discrete Toric Spaces

Toric spaces being non-simply connected, it is possible to find in such spaces some loops which are not homotopic to a point: we call them toric loops. Some applications, such as the study of the relationship between the geometrical characteristics of a material and its physical properties, rely on three-dimensional discrete toric spaces and require detecting objects having a toric loop. In this work, we study objects embedded in discrete toric spaces, and propose a new definition of loops and equivalence of loops. Moreover, we introduce a characteristic of loops that we call wrapping vector : relying on this notion, we propose a linear time algorithm which detects whether an object has a toric loop or not.