Representation of torus homeotopies by simple polyhedra with boundary

In 1991, Turaev and Viro constructed a quantum topological linear representation of mapping class groups of closed surfaces. To the mappings of a surface into itself, they assigned simple polyhedra whose boundaries consisted of two simple graphs cutting the surface into cells. The computational complexity of the Turaev-Viro representations strongly depends on the choice of suitable sets of simple polyhedra. In this paper, simple polyhedra for the torus are constructed. One of the reasons why they are convenient is that they all are obtained by gluing along boundary of copies of the same simple polyhedron.