TY  - JOUR
A1  - Kolbe, Benedikt Maximilian
A1  - Evans, Myfanwy E.
T1  - Isotopic tiling theory for hyperbolic surfaces
JF  - Geometriae dedicata
N2  - 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.
KW  - isotopic tiling theory
KW  - delaney-dress tiling theory
KW  - mapping class
KW  - groups
KW  - orbifolds
KW  - maps on surfaces
KW  - hyperbolic tilings
Y1  - 2020
U6  - https://doi.org/10.1007/s10711-020-00554-2
SN  - 0046-5755
SN  - 1572-9168
VL  - 212
IS  - 1
SP  - 177
EP  - 204
PB  - Springer
CY  - Dordrecht
ER  - 
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  - 
TY  - GEN
A1  - Kolbe, Benedikt Maximilian
A1  - Evans, Myfanwy E.
T1  - Isotopic tiling theory for hyperbolic surfaces
T2  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1347 
KW  - isotopic tiling theory
KW  - Delaney–Dress tiling theory
KW  - mapping class groups
KW  - Orbifolds
KW  - maps on surfaces
KW  - hyperbolic tilings
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-544285
SN  - 1866-8372
IS  - 1
ER  -