Related papers: Locally Helical Surfaces have bounded twisting
An unfolding of a polyhedron along its edges is called a vertex unfolding if adjacent faces are allowed to be connected at not only an edge but also a vertex. Demaine et al showed that every triangulated polyhedron has a vertex unfolding.…
We prove that any compact surface with constant positive curvature and conical singularities can be decomposed into irreducible components of standard shape, glued along geodesic arcs connecting conical singularities. This is a spherical…
Consider a 3$-$dimensional manifold $N$ obtained by gluing a finite number of ideal hyperbolic tetrahedra via isometries along their faces. By varying the isometry type of each tetrahedron but keeping fixed the gluing pattern we define a…
We consider triangulations of closed $2$-dimensional (not necessarily orientable) surfaces. Any minimal set of zigzags that double covers the set of edges provides a $z$-orientation of the triangulation. We introduce Markov chains of…
Understanding non-Haken 3-manifolds is central to many current endeavors in 3-manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox's theorem for submanifolds of the 3-sphere to…
The main objects of the paper are $z$-oriented triangulations of connected closed $2$-dimensional surfaces. A $z$-orientation of a map is a minimal collection of zigzags which double covers the set of edges. We have two possibilities for an…
We establish a correspondence between trisections of smooth, compact, oriented $4$--manifolds with connected boundary and diagrams describing these trisected $4$--manifolds. Such a diagram comes in the form of a compact, oriented surface…
In this paper, we explore minimal contact triangulations on contact 3-manifolds. We give many explicit examples of contact triangulations that are close to minimal ones. The main results of this article say that on any closed oriented…
In a recent paper we constructed a family of foliated 2-complexes of thin type whose typical leaves have two topological ends. Here we present simpler examples of such complexes that are, in addition, symmetric with respect to an involution…
We consider triangulations of closed surfaces S with a given set of vertices V; every triangulation can be branched that is enhanced to a Delta-complex. Branched triangulations are considered up to the b-transit equivalence generated by…
Transversal structures (also known as regular edge labelings) are combinatorial structures defined over 4-connected plane triangulations with quadrangular outer-face. They have been intensively studied and used for many applications…
Thin surfaces are ubiquitous in nature, from leaves to cell membranes, and in technology, from paper to corrugated containers. Structural thinness imbues them with flexibility, the ability to easily bend under light loads, even as their…
We characterize helix surfaces in the Berger sphere, that is surfaces which form a constant angle with the Hopf vector field. In particular, we show that, locally, a helix surface is determined by a suitable 1-parameter family of isometries…
We review Heisenberg homology of configurations in once bounded surfaces and extend the construction to the regular thickening of a finite graph with ribbon structure.
We provide a complete description of the edge-to-edge tilings with a regular triangle and a shield-shaped hexagon with no right angle. The case of a hexagon with a right angle is also briefly discussed.
Let $H_{g}$ be a 3-dimensional handlebody of genus $g$. We determine the twisted first homology group of the mapping class group of $H_{g}$ with coefficients in the first integral homology group of the boundary surface $\partial H_{g}$ for…
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…
In the present paper, a new type of ruled surfaces called osculating-type (OT)-ruled surface is introduced and studied. First, a new orthonormal frame is defined for OT-ruled surfaces. The Gaussian and the mean curvatures of these surfaces…
The topological dynamics of the horocyclic flow h_R on the unit tangent bundle of a geometrically finite hyperbolic surface is well known. In particular on such a surface the flow h_R is minimal or the minimal sets are the periodic orbits.…
P. Baird and the second author studied harmonic morphisms from a three-dimensional simply-connected space form to a surface and obtained a complete local and global classification of them. In this paper, we obtain a description of all…