Three configuration spaces in combinatorics

Roughly speaking, a reconfigurable system TZ is a discrete collection of positions of an object, along with local reversible moves. Such a system can be encoded as a cubical complex, which we call the configuration space S{1Z). When a configuration space is CAT(O), there is a unique shortest path between the vertices, and there is an efficient algorithm to compute this path. We can assess whether or not a cubical complex is CAT(O) by determining if there exists a corresponding poset with inconsistent pairs (PIP). In this thesis we show that the configuration spaces of the robotic arm in a rectangular tunnel Rm,n, of the tableaux T\ of hook shape A, and of the Dyck paths Dn of length 2n are CAT(O) cubical complexes. We do this by identifying the associated PIPs.