We present the mathematical background of a software package that computes triangulations of mapping tori of surface homeomorphisms, suitable for Jeff Weeks's program SnapPea. It consists of two programs. jmt computes triangulations and prints them in a human-readable format. jsnap converts this format into SnapPea's triangulation file format and may be of independent interest because it allows for quick and easy generation of input for SnapPea.

As an application, we obtain a new solution to the restricted conjugacy problem in the mapping class group.

Note that the above paragraph is correct only if we are already given a solution (via geometric techniques, say) to the isometry problem for hyperbolic three-manifolds. This inverts the standard way of thinking about the conjugacy problem but is not, despite appearences, totally crazy.

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