Related papers: $3$D Farey graph, lambda lengths and $SL_2$-tiling…

A meander of order n is a simple closed curve in the plane which intersects a horizontal line transversely at 2n points. (Meanders which differ by an isotopy of the line and plane are considered equivalent.) Let Gamma_n be the Cayley graph…

We study the structure of length three polynomial automorphisms of $R[X,Y]$ when $R$ is a UFD. These results are used to prove that if $\text{SL}_m(R[X_1,X_2,..., X_n]) = \text{E}_m(R[X_1,X_2,..., X_n])$ for all $n,\ge 0$ and for all $m \ge…

This paper develops a computational framework for studying the hyperbolic geometry of 2-bridge link complements and Kleinian groups generated by two parabolic elements. The framework is built on Sakuma-Weeks triangulations and introduces a…

In comparison to graphs, combinatorial methods for the isomorphism problem of finite groups are less developed than algebraic ones. To be able to investigate the descriptive complexity of finite groups and the group isomorphism problem, we…

In this paper, we give a characterization for a class of edge-transitive Cayley graphs, and provide methods for constructing Cayley graphs with certain symmetry properties. Also this study leads to construct and characterise a new family of…

We consider configurations of lines in 3-space with incidences prescribed by a graph. This defines a subvariety in a product of Grassmannians. Leveraging a connection with rigidity theory in the plane, for any graph, we determine the…

We study the Erdos distance problem over finite Euclidean and non-Euclidean spaces. Our main tools are graphs associated to finite Euclidean and non-Euclidean spaces that are considered in Bannai-Shimabukuro-Tanaka (2004, 2007). These…

We prove a generalized isoperimetric inequality for a domain diffeomorphic to a sphere that replaces filling volume with $k$-dilation. Suppose $U$ is an open set in $\mathbb{R}^n$ diffeomorphic to a Euclidean $n$-ball. We show that in…

Determining the number of (complex) realisations of a rigid graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we provide two new tools for determining realisation numbers in arbitrary…

Ribbon graphs embedded on a Riemann surface provide a useful way to describe the double line Feynman diagrams of large N computations and a variety of other QFT correlator and scattering amplitude calculations, e.g in MHV rules for…

An $SL_2$-tiling is a bi-infinite matrix in which all adjacent $2 \times 2$ minors are equal to $1$. Positive integral $SL_2$-tilings were introduced by Assem, Reutenauer and Smith as generalisations of classical Conway--Coxeter frieze…

We give a combinatorial description of the ``$D_{2n}$ planar algebra,'' by generators and relations. We explain how the generator interacts with the Temperley-Lieb braiding. This shows the previously known braiding on the even part extends…

In the CHY-frame for the tree-level amplitudes, the bi-adjoint scalar theory has played a fundamental role because it gives the on-shell Feynman diagrams for all other theories. Recently, an interesting generalization of the bi-adjoint…

Starting with topological field theories we investigate the Ray-Singer analytic torsion in three dimensions. For the lens Spaces L(p;q) an explicit analytic continuation of the appropriate zeta functions is contructed and implemented. Among…

We use Margulis' construction together with lattice counting arguments to build Cayley graphs on $\mathrm{SL}_{2}\left(\mathbb{F}_{p}\right),\;p\to\infty$ which are d-regular graphs with girth…

This paper studies when an arithmetical equivalence relation $E$ can be realized as the connectedness relation of a graph $G$ which is simpler to define than $E$. Several examples of such equivalence relations are established. In…

We study the long-standing open problem of efficiently testing rectilinear planarity of series-parallel graphs (SP-graphs) in the variable embedding setting. A key ingredient behind the design of a linear-time testing algorithm for…

Let $Y_{3,2}$ be the $3$-uniform hypergraph with two edges intersecting in two vertices. Our main result is that any $n$-vertex 3-uniform hypergraph with at least $\binom{n}{3} - \binom{n-m+1}{3} + o(n^3)$ edges contains a collection of $m$…

We prove an analogue of Belyi's theorem in characteristic two. Our proof consists of the following three steps. We first introduce a new notion called "pseudo-tame" for morphisms between curves over an algebraically closed field of…

Singerman introduced to the theory of maps on surfaces an object that is a universal cover for any map. This object is a tessellation of the hyperbolic plane together with a certain subset of the ideal boundary. The 1-skeleton of this…