arXiv research

New framework links fractal complexity to separation dimension.

This paper presents a novel scheme, based on a unique combination of genetic algorithms (GAs) and deep learning (DL), for the automatic reconstruction of Portuguese tile panels, a challenging real-world variant of the jigsaw puzzle problem (JPP) with important national heritage implications. Specifically, we introduce …

Investigates proving geometric theorems over complex and real numbers using tilings.

In this paper we prove the knight move theorem for the chromatic graph cohomologies with rational coefficients introduced by L. Helme-Guizon and Y. Rong. Namely, for a connected graph G with n vertices the only non-trivial cohomology groups $H^{i,n-i}(G)$, $H^{i,n-i-1}(G)$ come in isomorphic pairs: $H^{i,n-i}(G)\cong H…

Higher chromatic numbers $χ_s$ of simplicial complexes naturally generalize the chromatic number $χ_1$ of a graph. In any fixed dimension $d$, the $s$-chromatic number $χ_s$ of $d$-complexes can become arbitrarily large for $s\leq\lceil d/2\rceil$ [6,18]. In contrast, $χ_{d+1}=1$, and only little is known on $χ_s$ for …

A 2-complex requires at least 12 colours to avoid edge conflicts.

The paper reveals a property of chromatic homology for complete graphs.

The study of geometric group theory has suggested several theorems related to subdivision tilings that have a natural hyperbolic structure. However, few examples exist. We construct subdivision tilings for the complement of every nonsingular, prime alternating link. These tilings define a combinatorial space at infinit…

We study the connections between link invariants, the chromatic polynomial, geometric representations of models of statistical mechanics, and their common underlying algebraic structure. We establish a relation between several algebras and their associated combinatorial and topological quantities. In particular, we def…

Random square-tiled surfaces have normal genus distribution and cover all integer vectors.

In this paper we give a new characterization of the h-vector of the chromatic polynomial of a graph. We introduce reduced chromatic cohomology of a graph and show that h_i are its Betti numbers. We then discuss various combinatorial properties of these cohomologies. In particular we prove that these cohomologies depend…

This article is about chromatic numbers of hyperbolic surfaces. For a metric space, the $d$-chromatic number is the minimum number of colors needed to color the points of the space so that any two points at distance $d$ are of a different color. We prove upper bounds on the $d$-chromatic number of any hyperbolic surfac…

This paper introduces a conceptual framework, in the context of quantum topology and the algebras underlying it, for analyzing relations obeyed by the chromatic polynomial χ(Q) of planar graphs. Using it we give new proofs and substantially extend a number of classical results concerning the combinatorics of the chroma…

The Stanley chromatic symmetric function $X_G$ of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology of graded $S_n$-modules, whose graded Frobenius series $Frob_G(q,t)$ reduces to …

This paper studies closed 3-manifolds which are the attractors of a system of finitely many affine contractions that tile $\mathbb{R}^3$. Such attractors are called self-affine tiles. Effective characterization and recognition theorems for these 3-manifolds as well as theoretical generalizations of these results to hig…

The study proves that in normal tilings, at least two vertices are required per cell.

New tiles in higher dimensions are shown to be homeomorphic to balls.

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov h…

The paper defines new TQFTs from non-semisimple categories and proves spherical categories are chromatic.

The chromatic number of sphere graphs in 3-manifolds is bounded.

Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. We discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme-Guizon, Przy…

The traditional Riemann Mapping Theorem can be proved with circle packing techniques. We prove the Combinatorial Riemann Mapping Theorem for tilings of bounded size using circle packings.

Topology helps estimate chromatic numbers of random graphs on spheres.

Developed a new homology theory for graph chromatic polynomials.

In the first few homological gradings, there is an isomorphism between the Khovanov homology of a link and the categorification of the chromatic polynomial of a graph related to the link. In this article, we show that the categorification of the chromatic polynomial only contains torsion of order two, and hence Khovano…

Napoleon's theorem in elementary geometry describes how certain linear operations on plane polygons of arbitrary shape always produce regular polygons. More generally, certain triangulations of a polygon that tiles R^2 admit deformations which keep fixed the symmetry group of the tiling. This gives rise to isolation ph…

In this paper we show that the matrix of chromatic joins and the Gram matrix of the Temperley-Lieb algebra are similar (after rescaling), with the change of basis given by diagonal matrices.

We consider time-domain digital backpropagation with chromatic dispersion filters jointly optimized and quantized using machine-learning techniques. Compared to the baseline implementations, we show improved BER performance and >40% power dissipation reductions in 28-nm CMOS.

This paper defines girth for knots and links, linking it to Khovanov homology.

We study the chromatic number of the curve graph of a surface. We show that the chromatic number grows like k log k for the graph of separating curves on a surface of Euler characteristic -k. We also show that the graph of curves that represent a fixed non-zero homology class is uniquely t-colorable, where t denotes it…

For every orientable surface of finite negative Euler characteristic, we find a right-angled Artin group of cohomological dimension two which does not embed into the associated mapping class group. For a right-angled Artin group on a graph $\gam$ to embed into the mapping class group of a surface $S$, we show that the …

For each graph we construct graded cohomology groups whose graded Euler characteristic is the chromatic polynomial of the graph. We show the cohomology groups satisfy a long exact sequence which corresponds to the well-known deletion-contraction rule. This work is motivated by Khovanov's work on categorification of the…

The Penrose-Kauffman polynomial connects knot theory to graph coloring.

The hyperbolic structure of equilateral pentagons is mapped to a tiling of the hyperbolic plane.

This note characterizes monohedral tilings of regular polygons with up to three tiles.

The study of tiling homology on flat surfaces, proving impossibility of certain tilings.

New tiles allow efficient knot mosaics for small knots.

Rep-tiles fill cubes in any dimension.

Study tiling spaces over irrational tori using diffeological classification.

Motivated by the work in [15], this paper deals with the theory of the braids from chromatic configuration spaces. This kind of braids possess the property that some strings of each braid may intersect together and can also be untangled, so they are quite different from the ordinary braids in the sense of Artin. This e…

Shellable tilings on simplicial complexes help understand their structure.

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 relative chromatic number $c\_0(S)$ of a compact surface $S$ with boundary is defined as the supremum of the chromatic numbers of graphs embedded in $S$ with all vertices on $\partial S$. This topological invariant was introduced for the study of the multiplicity of the first Steklov eigenvalue of $S$. In this arti…

The study classifies tilings of the sphere by congruent quadrilaterals.

New method constructs tilings of the plane using directed edges and alignments.

New tile types for knots and links reduce complexity.

Study on tilings of the plane with two types of tiles of varying areas.

New spectral sequences derived from shellable tilings.