Enumerating Isotopy Classes of Tilings Guided by the Symmetry of Triply Periodic Minimal Surfaces

Kolbe, Benedikt; Evans, Myfanwy E.

doi:10.1137/20m1358943

Cited by 4 publications

(4 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, tessellations of H 2 can be realised by a broad variety of finite tiles, whose edges induce crystalline networks, of relevance to chemistry, physics, and mathematics. [32][33][34][35][36][37][38][39][40][41] Further, tessellations of H 2 by infinite ribbon-shaped tiles, related to so-called ''free tilings'' 34,35,42,43 map to complex interwoven networks in E 3 via TPMS, which match patterns found in numerical simulations of selfassembled star polymers. 42,44,45 The latter will be reconsidered later in this paper.…”

Section: Introductionsupporting

confidence: 66%

Mapping hyperbolic order in curved materials

et al. 2023

View full text Add to dashboard Cite

show abstract

Section: Introductionsupporting

confidence: 66%

Mapping hyperbolic order in curved materials

et al. 2023

View full text Add to dashboard Cite

show abstract

“…Returning to the initial inspiration of this work -the question of how to enumerate stripe patterns on the gyroid-this can now be achieved by finding the (branched) ribbon tilings in the symmetry groups compatible with the covering map that wraps the hyperbolic plane onto this periodic surface. Moreover, since we fix the symmetry group of the tilings under investigation, the methods of [28,27] give a natural approach to using the theory developed here for enumerations and investigations into isotopy classes of ribbon tilings on non-simply connected surfaces.…”

Section: Discussionmentioning

confidence: 99%

“…The exploration of crystallographic line and tree patterns in the hyperbolic plane goes back to Hyde et al [22,13,14,12]. Our approach to enumeration using graphs on orbifolds and D-symbols allows us to adapt the methods of isotopic tiling theory [28,27] to systematically enumerate the distinct ways in which tilings fit on compact surfaces.…”

Section: Introductionmentioning

confidence: 99%

Tile-Transitive Tilings of the Euclidean and Hyperbolic Planes by Ribbons

Kolbe

Robins

2022

Association for Women in Mathematics Series

Self Cite

View full text Add to dashboard Cite

We present a method to enumerate tile-transitive crystallographic tilings of the Euclidean and hyperbolic planes by unbounded ribbon tiles up to equivariant equivalence. The hyperbolic case is relevant to self-assembly of branched polymers. This is achieved by combining and extending known methods for enumerating crystallographic disk-like tilings. We obtain a natural way of describing all possible stabiliser subgroups of tile-transitive tilings using a topological viewpoint of the tile edges as a graph embedded in an orbifold, and a group theoretical one derived from the structure of fundamental domains for discrete groups of planar isometries.

show abstract

“…PBCs are often employed to model physical systems of filaments in order to avoid boundary effects. The entanglement in such systems has several characteristics that make it more difficult to quantify [5,10,[22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. A system with PBC is created by infinite copies of a generating cell which creates an infinite system whose collective geometric/topological entanglement is impossible to compute as a whole.…”

Section: Introductionmentioning

confidence: 99%

The Jones polynomial in systems with periodic boundary conditions

Barkataki,

Panagiotou

2024

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Entanglement of collections of ﬁlaments arises in many contexts, such as in polymer melts, textiles and crystals. Such systems are modeled using periodic boundary conditions (PBC), which create an inﬁnite periodic system whose global entanglement may be impossible to capture and is repetitive. We introduce two new methods to assess topological entanglement in PBC: the Periodic Jones polynomial and the Cell Jones polynomial. These tools capture the grain of entanglement in a periodic system of open or closed chains, by using a ﬁnite link as a representative of the global system. These polynomials are topological invariants in some cases, but in general are sensitive to both the topology and the geometry of physical systems. For a general system of 1 closed chain in 1 PBC, we prove that the Periodic Jones polynomial is a recurring factor, up to a remainder, of the Jones polynomial of a conveniently chosen ﬁnite cutoﬀ of arbitrary size of the inﬁnite periodic system. We apply the Cell Jones polynomial and the Periodic Jones polynomial to physical PBC systems such as 3D realizations of textile motifs and polymer melts of linear chains obtained frommolecular dynamics simulations. Our results demonstrate that the Cell Jones polynomial and the Periodic Jones polynomial can measure collective entanglement complexity in such systems of physical relevance.

show abstract

Enumerating Isotopy Classes of Tilings Guided by the Symmetry of Triply Periodic Minimal Surfaces

Cited by 4 publications

References 48 publications

Mapping hyperbolic order in curved materials

Mapping hyperbolic order in curved materials

Tile-Transitive Tilings of the Euclidean and Hyperbolic Planes by Ribbons

The Jones polynomial in systems with periodic boundary conditions

Contact Info

Product

Resources

About