arXiv research

We consider the solid angle that a planar compact subset subtends at a point in a level set of height h and study two extremal problems for the solid angle. One of the variables is a point in such a plane, that is, we study the properties of the solid angle maximizer. The other is the pair of a planar compact subset an…

This note generalizes the visual angle to convex sets in 3D space.

This paper explores historical and philosophical aspects of angles and solid angles, inspired by Euler's work.

The paper introduces surfaces with constant solid angle for designing shell structures.

Given a smooth closed oriented manifold $M$ of dimension $n$ embedded in $\mathbb{R}^{n+2}$ we study properties of the `solid angle' function $Φ\colon\mathbb{R}^{n+2}\setminus M\to S^1$. It turns out that a non-critical level set of $Φ$ is an explicit Seifert hypersurface for $M$.

Researchers solved a geometry paradox for creased tubes.

We describe the deformation space of a solid torus with boundary modelled on convex ideal hyperbolic polyhedra. This deformation space is given by natural Gauss--Bonnet type inequalities on the dihedral angles. The result extends to solid tori with an arbitrary conical singularity along the core. Our method is to decom…

Minimal surfaces span periodic curves in 3D space.

In this paper we investigate constant mean curvature surfaces with nonempty boundary in Euclidean space that meet a right cylinder at a constant angle along the boundary. If the surface lies inside of the cylinder, we obtain some results of symmetry by using the Alexandrov reflection method. When the mean curvature is …

In this work we simulate null geodesics for the Bonnor massive dipole metric by implementing a symbolic-numerical algorithm in Sage and Python. This program is also capable of visualizing in 3D, in principle, the geodesics for any given metric. Geodesics are launched from a common point, collectively forming a cone of …

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle. We study under what geometric conditions the surface must …

We show that a capillary surface in a solid cone, that is, a surface that has constant mean curvature and the boundary of surface meets the boundary of the cone with a constant angle, is radially graphical if the mean curvature is non-positive with respect to the Gauss map pointing toward the domain bounded by the surf…

Differentiable voxelization for 3D meshes with GPU acceleration.

Euler explored spherical geometry using trigonometric formulae and solid geometry methods.

The paper proves a Willmore-type inequality for unbounded convex sets.

We study the 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry. For a triangulation of a 3-manifold, we prove that if the number of tetrahedra incident to each vertex is at least 23, then there exist real or virtual ball packings with vanishing (extended) combinatorial scalar curvature, i.e. the…

Identifies Heegaard Floer homology solid tori via Dehn fillings.

Proves NP and co-NP status for knot core recognition in solid torus.

New theorem for 4D links simplifies characterisation problem.

Classifies small links in an unmarked solid torus.

We prove that for every smooth Jordan curve $γ$, if $X$ is the set of all $r \in [0,1]$ so that there is an inscribed rectangle in $γ$ of aspect ratio $\tan(r\cdot π/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedde…

New method constructs Seifert solids from bridge trisections.

We show that in any triangulation of a solid torus, there is a pre-core curve that lies in the 2-skeleton and that intersects the interior of each face in at most 10 straight arcs. By definition, a pre-core curve is a simple closed curve that becomes a core curve when a collar is attached to the boundary of the solid t…

The paper classifies all tight contact structures on a solid torus.

Study Murasugi sum in 4D for knotted surfaces, defining arborescent surfaces.

Geometrically reformulates Cosserat solid mechanics using differential geometry.

We interpret Coxeter's truncated braid groups in terms of Platonic solids.

Normalizing flows model atomic solids without needing ground-truth samples.

The paper evaluates homology for links in a solid torus with special boundary conditions.

Researchers describe how special conic bundles deform into double solids.

We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diag…

We use the topological invariant of spatial graphs introduced by S. Yamada to find necessary conditions for a spatial graph to be periodic with a prime period. The proof of the main result is based on computing the Yamada skein algebra of the solid torus then proving that this algebra injects into the Kauffman bracket …

Formula connects knot invariant to Lefschetz number, proving special case for Seifert solids.

Extends Frohman and Rannard's result to Seifert fiber spaces with singular surfaces.

Maps from rational homology solid tori yield rank inequalities in Heegaard Floer homology.

New method fractures hyperbolic manifolds using cone singularities.

We construct new embedded self-shrinkers of genus 3, 5, 7, 11 and 19 using variational methods. Our self-shrinkers resemble doublings of the Platonic solids and were discovered numerically by D. Chopp in 1994.

Study on knots in contact manifolds, focusing on their width and thickness.

We use the 3d-3d correspondence together with the DGG construction of theories $T_n[M]$ labelled by 3-manifolds M to define a non-perturbative state-integral model for SL(n,C) Chern-Simons theory at any level k, based on ideal triangulations. The resulting partition functions generalize a widely studied k=1 state-integ…

We present explicit geometric decompositions of the complement of tiling links, which are alternating links whose projection graphs are uniform tilings of the 2-sphere, the Euclidean plane or the hyperbolic plane. This requires generalizing the angle structures program of Casson and Rivin for triangulations with a mixt…

Skeleta of Platonic solids are factored into spheres.

New periodic polyhedra found in curved spaces.

Researchers compute $\mathfrak{gl}_2$-skein modules for lens spaces.

DM approximates submanifolds with error bounds.

Let $K= K(w,b,t)$ be a 1-bridge braid in a solid torus $V$, and let $γ$ be a $(p,q)$ curve on the torus $T = \partial V$ of the exterior $M_K$ of $K$. It will be shown that Dehn filling on $T$ along $γ$ produces a solid torus if and only if $p$ and $q$ satisfy one of four conditions determined by the parameters $(w,b,t…

The present paper is devoted to the joint motion of two immiscible incompressible liquids in porous media. The liquids have different densities and initially separated by a surface of strong discontinuity (free boundary). We discuss the results of numerical simulations for exact free boundary problems on the microscopi…

A vertex-transitive map $X$ is a map on a surface on which the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is …

In this paper, we develop a method for constructing left-orders on the fundamental groups of rational homology 3-spheres. We begin by constructing the holonomy extension locus of a rational homology solid torus $M$, which encodes the information about peripherally hyperbolic $\widetilde{\text{PSL}_2\mathbb{R}}$ represe…