Related papers: Quasiperiodic bobbin lace patterns
This paper introduces a new systematic algorithm for constructing periodic Euclidean weaving diagrams with combinatorial arguments. It is shown that such a weaving diagram can be considered as a specific type of four-regular periodic planar…
A near-group category is an additively semisimple category with a product such that all but one of the simple objects is invertible. We classify braided structures on near-group categories, and give explicit numerical formulas for their…
Aperiodic algebras are infinite dimensional algebras with generators corresponding to an element of the aperiodic set. These algebras proved to be an useful tool in studying elementary excitations that can propagate in multilayered…
A knotted ribbon is one of physical aspect of a knot. A folded ribbon knot is a depiction of a knot obtained by folding a long and thin rectangular strip to become flat. The ribbonlength of a knot type can be defined as the minimum length…
Aperiodic tilings are non-periodic tilings characterized by local constraints. They play a key role in the proof of the undecidability of the domino problem (1964) and naturally model quasicrystals (discovered in 1982). A central question…
Quasicrystals are aperiodically ordered solids that exhibit long-range order without translational periodicity, bridging the gap between crystalline and amorphous materials. Due to their lack of translational periodicity, information on…
The generation of high-order harmonics in finite, hexagonal nanoribbons is simulated. Ribbons with armchair and zig-zag edges are investigated by using a tight-binding approach with only nearest neighbor hopping. By turning an alternating…
The trilobite and crab are among the very simplest aperiodic sets of tiles known: two tiles in eight translation classes. Yet the proof that they are an aperiodic set is surprisingly complex.
In this note we show that ribbon concordance forms a partial ordering on the set of knots, answering a question of Gordon. The proof makes use of representation varieties of the knot groups to $SO(N)$ and relations between them induced by a…
The authors conjectured previously that a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove the conjecture for a class of knots that includes all knots of genus 1, using techniques from…
We study the electronic transport in quasiperiodic separable tight-binding models in one, two, and three dimensions. First, we investigate a one-dimensional quasiperiodic chain, in which the atoms are coupled by weak and strong bonds…
This is a review article on the Bennequin-Birman-Menasco machinery for studying embedded incompressible surfaces in 3-space via their `braid foliations'. Two cases are investigated: case (1) The surface has non-empty boundary; the boundary…
Tiling models can reveal unexpected ways in which local constraints give rise to exotic long-range spatial structure. The recently discovered Hat monotile (and its mirror image) has been shown to be aperiodic~[Smith et al., arXiv:2303.10798…
Curve stitching is a classic educational activity where one constructs elegant curves from a family of straight lines. We perform curve stitching around a circle to make a modular stitch graph. Take $m$ points equally spaced around a…
Quasicrystals lack translational symmetry, but can still exhibit long-ranged order, promoting them to candidates for unconventional physics beyond the paradigm of crystals. Here, we apply a real-space functional renormalization group…
We introduce a novel combinatorial structure called pointed building sets, which can be viewed as families of lattices equipped with compatibility relations. To each pointed building set $\mathsf{B}$, we associate a complete lattice…
Categorical skew lattices are a variety of skew lattices on which the natural partial order is especially well behaved. While most skew lattices of interest are categorical, not all are. They are characterized by a countable family of…
We consider the problem of finding quasiperiodicities in a Fibonacci string. A factor u of a string y is a cover of y if every letter of y falls within some occurrence of u in y. A string v is a seed of y, if it is a cover of a superstring…
Some curious structural similarities between a recent braid- and Hurwitz algebraic description of the unbroken internal symmetries for a single generations of Standard Model fermions were recently identified. The non-trivial braid groups…
We introduce semidistrim lattices, a simultaneous generalization of semidistributive and trim lattices that preserves many of their common properties. We prove that the elements of a semidistrim lattice correspond to the independent sets in…