A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
New tile types for knots and links reduce complexity.
Paper proves corner connection tiles can represent knots with fewer tiles.
We study square-tiled tori, that is, tori obtained from a finite collection of unit squares by parallel side identifications. Square-tiled tori can be parametrized in a natural way that allows to count the number of square-tiled tori tiled by a given number of square tiles. There is a natural $\mathrm{SL}(2,\mathbf{Z})…
New tiles allow efficient knot mosaics for small knots.
In this paper we introduce the concept of a space-efficient knot mosaic. That is, we seek to determine how to create knot mosaics using the least number of non-blank tiles necessary to depict the knot. This least number is called the tile number of the knot. We determine strict bounds for the tile number of a knot in t…
Softens tilings in 3D space, proving conjectures about polyhedral tilings.
Paper introduces spherical knot mosaics for knot and link invariants.
A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main result is explicit formulas for the weighted number of convex tilings with a given number of tiles. To prove these for…
Shellable tilings on simplicial complexes help understand their structure.
In 2008, Kauffman and Lomonaco introduce the concepts of a knot mosaic and the mosaic number of a knot or link, the smallest integer such that a knot or link can be represented on an -mosaic. In arXiv:1702.06462, the authors explore space-efficient knot mosaics and the tile number of a knot or link, the smallest…
Investigates proving geometric theorems over complex and real numbers using tilings.
This note characterizes monohedral tilings of regular polygons with up to three tiles.
Algorithm finds mosaic numbers for knots with 10 or fewer crossings.
The study explores maps of 2- and 3-uniform tilings on the torus.
In this note we prove that any monohedral tiling of the closed circular unit disc with topological discs as tiles has a -fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain t…
Constructs surfaces that can be tiled by a finite set of rigid motion congruence classes of tiles.
Study on tilings of the plane with two types of tiles of varying areas.
New method constructs tilings of the plane using directed edges and alignments.
Several articles deal with tilings with squares and dominoes on 2-dimensional boards, but only a few on boards in 3-dimensional space. We examine a tiling problem with colored cubes and bricks of -board in three dimensions. After a short introduction and the definition of breakability we show a way …
The main goal of this paper is to define a 1-1 correspondence between between substitution tilings constructed by inflation and the arithmetic of positional representation in the underlying real vector space. It introduces a generalization of inflationary tessellations to equivalence classes of tiles. Two tiles belong …
The concepts of tile number and space-efficiency for knot mosaics were first explored by Heap and Knowles (arXiv:1702.06462), where they determined the possible tile numbers and space-efficient layouts for every prime knot with mosaic number 6 or less. In this paper, we extend those results to prime knots with mosaic n…
Study of random multicurves and square-tiled surfaces on large genus surfaces.
Soft cells fill space without gaps, derived from minimal surfaces and deformed using edge bending.
We develop a recursive formula for counting the number of rectangulations of a square, i.e the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations, as analyzed by Reading in [5]. Our computations agree with [5] as far as was ca…
A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, …
The study proves that in normal tilings, at least two vertices are required per cell.
We call "flippable tilings" of a constant curvature surface a tiling by "black" and "white" faces, so that each edge is adjacent to two black and two white faces (one of each on each side), the black face is forward on the right side and backward on the left side, and it is possible to "flip" the tiling by pushing all …
Characterizes arithmetic and commensurable links in curved surfaces.
The study finds the bounds of vertex orbits in maps derived from specific lattices.
If the face-cycles at all the vertices in a map on a surface are of same type then the map is called semi-equivelar. There are eleven types of Archimedean tilings on the plane. All the Archimedean tilings are semi-equivelar maps. If a map on the torus is a quotient of an Archimedean tiling on the plane then the map…
In every connected component of every stratum of Abelian differentials, we construct square-tiled surfaces with one vertical and one horizontal cylinder. We show that for all but the hyperelliptic components this can be achieved in the minimum number of squares necessary for a square-tiled surface in that stratum. For …
Estimates surface count with prescribed foliations.
The study of tiling homology on flat surfaces, proving impossibility of certain tilings.
Researchers found the maximum number of holes in polyominoes grows proportionally to the dimension.
Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length generically has quadratic asymptotics in this length, with a common coefficient c…
Rep-tiles fill cubes in any dimension.
Study tiling spaces over irrational tori using diffeological classification.
In this article we study Ammann tilings from the perspective of symplectic geometry. Ammann tilings are nonperiodic tilings that are related to quasicrystals with icosahedral symmetry. We associate to each Ammann tiling two explicitly constructed highly singular symplectic spaces and we show that they are diffeomorphic…
The study classifies tilings of the sphere by congruent quadrilaterals.
New spectral sequences derived from shellable tilings.
We determine the topology of the moduli space of periodic tilings of the plane by parallelograms. To each such tiling, we associate combinatorial data via the zone curves of the tiling. We show that all tilings with the same combinatorial data form an open subset in a suitable Euclidean space that is homotopy equivalen…
This paper classifies all 3D rep-tiles up to homeomorphism.
Many tasks require finding groups of elements in a matrix of numbers, symbols or class likelihoods. One approach is to use efficient bi- or tri-linear factorization techniques including PCA, ICA, sparse matrix factorization and plaid analysis. These techniques are not appropriate when addition and multiplication of mat…
4-ball can be tiled with knotted surfaces.
We describe a method to classify crystallographic tilings of the Euclidean and hyperbolic planes by tiles whose stabiliser group contains translation isometries or whose topology is not that of a closed disk. We tackle this problem from two different viewpoints, one with constructive techniques to enumerate such tiling…
Study higher rank inner products and their tilings to describe tori degenerations.
In this paper we describe the pentagonal tiling of the plane defined in the article "A regular pentagonal tiling of the plane" by P. L. Bowers and K. Stephenson as a conformal substitution tiling and summarize many of its properties given in the mentioned article. We show furthermore why such tiling is not FLC with res…
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 …