 
Summary: GEODESICS IN CAT(0) CUBICAL COMPLEXES
FEDERICO ARDILA, MEGAN OWEN, AND SETH SULLIVANT
Abstract. We describe an algorithm to compute the geodesics in an arbitrary CAT(0)
cubical complex. A key tool is a correspondence between cubical complexes of global
nonpositive curvature and posets with inconsistent pairs. This correspondence also gives
an explicit realization of such a complex as the state complex of a reconfigurable system,
and a way to embed any interval in the integer lattice cubing of its dimension.
1. Introduction
A cubical complex is a polyhedral complex where all cells are cubes and all attaching
maps are injective. Informally speaking, it is just like a simplicial complex, except that
the cells are cubes instead of simplices. Every cubical complex has an intrinsic metric
induced by the Euclidean L2
metric on each cube. A polyhedral complex is CAT(0) if
and only if it is globally nonpositively curved. This implies that there is a unique local
geodesic between any two points. CAT(0) cubical complexes make frequent appearances
in mathematics and its applications, for instance in geometric group theory, in the theory
of reconfigurable systems, and in phylogenetics. The main goal of this paper is to describe
an algorithm for computing geodesics in CAT(0) cubical complexes.
A prototypical example of CAT(0) cubical complexes comes from "reconfigurable sys
tems", a broad family of systems which change according to local rules. Examples include
