TY  - JOUR
A1  - Kolbe, Benedikt Maximilian
A1  - Evans, Myfanwy E.
T1  - Enumerating isotopy classes of tilings guided by the symmetry of triply
JF  - Siam journal on applied algebra and geometry
N2  - 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.
KW  - isotopic tiling theory
KW  - mapping class group
KW  - orbifolds
KW  - group
KW  - presentations
KW  - representations of groups as automorphism groups of
KW  - algebraic systems
KW  - triply periodic minimal surface
KW  - Delaney--Dress
KW  - tiling theory
KW  - hyperbolic tilings
KW  - two-dimensional topology
Y1  - 2022
U6  - https://doi.org/10.1137/20M1358943
SN  - 2470-6566
VL  - 6
IS  - 1
SP  - 1
EP  - 40
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  -