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. The package is an extension of the software described in Brinkmann . 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.
"Computing Triangulations of Mapping Tori of Surface Homeomorphisms." Experiment. Math. 10 (4) 571 - 582, 2001.