  We consider hyperbolic structures on the compression body C with genus 2
positive boundary and genus 1 negative boundary. Note that C deformation
retracts to the union of the torus boundary and a single arc with its endpoints
on the torus. We call this arc the core tunnel of C. We conjecture that, in any
geometrically finite structure on C, the core tunnel is isotopic to a geodesic.
By considering Ford domains, we show this conjecture holds for many
geometrically finite structures. Additionally, we give an algorithm to compute
the Ford domain of such a manifold, and a procedure which has been implemented
to visualize many of these Ford domains. Our computer implementation gives
further evidence for the conjecture.
