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We present a technique for the enumeration of all isotopically distinct waysof tiling a hyperbolic surface of finite genus, possibly nonorientable and withpunctures and boundary. This provides a generalization of the enumeration ofDelaney-Dress combinatorial tiling theory on the basis of isotopic tilingtheory. To accomplish this, we derive representations of the mapping classgroup of the orbifold associated to the symmetry group in the group of outerautomorphisms of the symmetry group of a tiling. We explicitly givedescriptions of certain subgroups of mapping class groups and of tilings asdecorations on orbifolds, namely those that are commensurate with thePrimitive, Diamond and Gyroid triply-periodic minimal surfaces. We use thisexplicit description to give an array of examples of isotopically distincttilings of the hyperbolic plane with symmetries generated by rotations,outlining how the approach yields an unambiguous enumeration.

Enumerating Isotopy Classes of Tilings guided by the symmetry of Triply-Periodic Minimal Surfaces