@article{KolbeEvans2020,
  author    = {Kolbe, Benedikt Maximilian and Evans, Myfanwy E.},
  title     = {Isotopic tiling theory for hyperbolic surfaces},
  series = {Geometriae dedicata},
  volume    = {212},
  journal   = {Geometriae dedicata},
  number    = {1},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {0046-5755},
  doi       = {10.1007/s10711-020-00554-2},
  pages     = {177 -- 204},
  year      = {2020},
  abstract  = {In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.},
  language  = {en}
}
@article{KolbeEvans2022,
  author    = {Kolbe, Benedikt Maximilian and Evans, Myfanwy E.},
  title     = {Enumerating isotopy classes of tilings guided by the symmetry of triply},
  series = {Siam journal on applied algebra and geometry},
  volume    = {6},
  journal   = {Siam journal on applied algebra and geometry},
  number    = {1},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {2470-6566},
  doi       = {10.1137/20M1358943},
  pages     = {1 -- 40},
  year      = {2022},
  abstract  = {We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This generalizes the enumeration using Delaney--Dress combinatorial tiling theory of combinatorial classes of tilings to isotopy classes of tilings. To accomplish this, we derive an action of the mapping class group of the orbifold associated to the symmetry group of a tiling on the set of tilings. We explicitly give descriptions and presentations of semipure mapping class groups and of tilings as decorations on orbifolds. We apply this enumerative result to generate an array of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the threedimensional symmetries of the primitive, diamond, and gyroid triply periodic minimal surfaces, which have relevance to a variety of physical systems.},
  language  = {en}
}