  We consider the problem of optimal path planning in different homotopy
classes in a given environment. Though important in robotics applications,
path-planning with reasoning about homotopy classes of trajectories has
typically focused on subsets of the Euclidean plane in the robotics literature.
The problem of finding optimal trajectories in different homotopy classes in
more general configuration spaces (or even characterizing the homotopy classes
of such trajectories) can be difficult. In this paper we propose automated
solutions to this problem in several general classes of configuration spaces by
constructing presentations of fundamental groups and giving algorithms for
solving the \emph{word problem} in such groups. We present explicit results
that apply to knot and link complements in 3-space, discuss how to extend to
cylindrically-deleted coordination spaces of arbitrary dimension, and also
present results in the coordination space of robots navigating on an Euclidean
plane.
