Related papers: Characterisation of geodesic self-dual regular sur…
We study arrangements of geodesic arcs on a sphere, where all arcs are internally disjoint and each arc has its endpoints located within the interior of other arcs. We establish fundamental results concerning the minimum number of arcs in…
The concept of a normal surface in a triangulated, compact 3-manifold was generalised by Thurston to a spun-normal surface in a non-compact 3-manifold with ideal triangulation. This paper defines a boundary curve map which takes a…
We define a class a metric spaces we call Brownian surfaces, arising as the scaling limits of random maps on general orientable surfaces with a boundary and we study the geodesics from a uniformly chosen random point. These metric spaces…
The question of whether a closed Riemannian manifold has infinitely many geometrically distinct closed geodesics has a long history. Though unsolved in general, it is well understood in the case of surfaces. For surfaces of revolution…
A model of simplicial quantum gravity in three dimensions(3D) was investigated numerically based on the technique of dynamical triangulation (DT). We are concerned with the genus of surfaces appearing on boundaries (i.e., sections) of a 3D…
Let $S$ be a hyperbolic oriented Riemann surface of finite type. The main purpose of this paper is to show that non-trivial geometric intersection between closed curves on $S$ is detected by some symplectic submodules they naturally…
With the $[0,1,2]$-family of cyclic triangulations we introduce a rich class of vertex-transitive triangulations of surfaces. In particular, there are infinite series of cyclic $q$-equivelar triangulations of orientable and non-orientable…
We consider the rational linear relations between real numbers whose squared trigonometric functions have rational values, angles we call ``geodetic''. We construct a convenient basis for the vector space over Q generated by these angles.…
This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under…
We will consider a two dimensional "symmetric" subfamily of the four dimensional family of Fricke cubic surfaces. The main result is that such symmetric cubic surfaces arise as character varieties for the exceptional group of type G_2.…
The triangulations of a surface $\Sigma$ with a prescribed set of vertices can be endowed with a graph structure $\mathcal{F}(\Sigma)$. Its edges connect two triangulations that differ by a single arc. It is known that, when $\Sigma$ is a…
A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of…
Gordon and Wilton recently proved that the double D of a free group F amalgamated along a cyclic subgroup C of F contains a surface group if a generator w of C satisfies a certain 3-manifold theoretic condition, called virtually…
Let $\Delta$ be a $g_2$-minimal normal 3-pseudomanifold. A vertex in $\Delta$ whose link is not a sphere is called a singular vertex. When $\Delta$ contains at most two singular vertices, its combinatorial characterization is known [9]. In…
We define a 2-normal surface to be one which intersects every 3-simplex of a triangulated 3-manifold in normal triangles and quadrilaterals, with one or two exceptions. The possible exceptions are a pair of octagons, a pair of unknotted…
We study the geodesics on an invariant surface of a three dimensional Riemannian manifold. The main results are: the characterization of geodesic orbits; a Clairaut's relation and its geometric interpretation in some remarkable three…
Homological mirror symmetry predicts that there is a relation between autoequivalence groups of derived categories of coherent sheaves on Calabi-Yau varieties, and the symplectic mapping class groups of symplectic manifolds. In this paper,…
A $\textit{regular polygon surface}$ $M$ is a surface graph $(\Sigma, \Gamma)$ together with a continuous map $\psi$ from $\Sigma$ into Euclidean 3-space which maps faces to regular Euclidean polygons. When $\Sigma$ is homeomorphic to the…
Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but…
We discuss self-crossing patterns of closed geodesics on a convex surface.