arXiv research

Study of elementary planes in Apollonian orbifold with unusual equidistribution.

Two new proofs classify complete totally geodesic subsets of complex hyperbolic plane.

We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz-Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces.

We consider non-elementary Kleinian groups Γ, without invariant plane, generated by an elliptic and a hyperbolic element with their axes lying in one plane. We find presentations and a complete list of orbifolds uniformized by such Γ.

The paper uses complex-valued functions to simplify plane differential geometry and kinematics.

We give a complete list of orbifolds uniformised by discrete non-elementary two-generator subgroups of PSL(2,C) without invariant plane whose generators and their commutator have real traces.

Study shows no smooth embeddings of rational homology balls into complex projective plane.

The aim of this text is to provide an elementary and self-contained exposition of Gromov's argument on topological overlap (the presentation is based on Gromov's work, as well as two follow-up papers of Matousek and Wagner, and of Dotterrer, Kaufman and Wagner). We also discuss a simple generalization in which the vert…

A short, elementary proof is given of the result: The number of components of a link arising from a medial graph M(G) by resolving vertices is equal to the nullity of the mod-2 Laplacian matrix of the graph G.

New proof shows Goldberg's kernel is not finitely generated.

We study the problem of rigidity of closures of totally geodesic plane immersions in geometrically finite manifolds containing rank $1$ cusps. We show that the key notion of K-thick recurrence of horocycles fails generically in this setting. This property was introduced in the recent work of McMullen, Mohammadi and Oh.…

New moves for singular knots identified and described.

In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index $b(L)$ and arc index $α(L)$ for any knot or non-split link $L$, which are $b(L) \leq \frac{1}{2} c(L) +…

The asymptotic behavior of open plane sections of triply periodic surfaces is dictated, for an open dense set of plane directions, by an integer second homology class of the three-torus. The dependence of this homology class on the direction can have a rather rich structure, leading in special cases to a fractal. In th…

The paper defines invariants for almost graph embeddings and explores their properties.

For an arrangement of $n$ pseudolines in the real projective plane let us denote by $t_i$ the number of vertices incident to $i$ lines. We obtain a linear on $t_i$ inequality similar to the Hirzebruch one, but with an elementary proof. We present an algorithm for producing lower bounds of the number of regions basing o…

We consider congruences of straight lines in a plane with the combinatorics of the square grid, with all elementary quadrilaterals possessing an incircle. It is shown that all the vertices of such nets (we call them incircular or IC-nets) lie on confocal conics. Our main new results are on checkerboard IC-nets in the p…

We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and combinatorics. We define a mod2-valued self-intersection invariant (i.e. the van Kampe…

Proof of Thurston's earthquake theorem using Anti-de Sitter geometry.

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…

The paper explores invariants of graph drawings in the plane.

There are three main thrusts to this article: a new proof of Levi's Enlargement Lemma for pseudoline arrangements in the real projective plane; a new characterization of pseudolinear drawings of the complete graph; and proofs that pseudolinear and convex drawings of $K_n$ have $n^2+{}$O$(n\log n)$ and O$(n^2)$, respect…

We define an elementary relatively $\mathbb Z/4$ graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram obtained by taking traceless $SU(2)$ character varieties of 2-tangle decompositions of knots. Calculations for torus knots…

The purpose of this article is to \begin{enumerate} \item define $M(t,k)$ the $t$-fold center of mass arrangement for $k$ points in the plane, \item give elementary properties of $M(t,k)$ and \item give consequences concerning the space $M(2,k)$ of $k$ distinct points in the plane, no four of which are the vertices of …

Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either…

An elementary geometric construction is used to relate the space of lattices in a plane to the space exp_3(S^1) of the subsets of a circle of cardinality at most 3. As a consequence we obtain new proofs of a theorem of Bott which says that exp_3(S^1) is homeomorphic to a 3-sphere and a theorem of Shchepin which says th…

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

The paper proves geodesics and conic sections are length-minimizing under specific metrics.

Here we study the deformations of associative submanifolds inside a G_2 manifold M^7 with a calibration 3-form φ. A choice of 2-plane field Λon M (which always exits) splits the tangent bundle of M as a direct sum of a 3-dimensional associate bundle and a complex 4-plane bundle TM= E\oplus V, and this helps us to relat…

This note is purely expository. A subset N of the plane is affine ambient homogeneous if for each x,y in N there exists an affine transformation taking x to y and N to itself. The result of D. Repovs, E. V. Scepin and the author on such subsets is presented, together with discussion, corollaries and generalizations. At…

We obtain an upper bound for the volume of the convex hull of a simple closed Frenet curve with exactly four vertices, i.e., four points of vanishing torsion, and lying on the boundary of its convex hull. Moreover, we show that the upper bound is attained when the curve intersects every plane in at most four points, a …

Derives hyperbolic laws of cosines and sines with fermionic corrections.

Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in $\math…

Abstract: Study of geometric structures on surfaces using various tools.

The paper constructs quantum invariants for knotoid diagrams.

The classical Björling problem is to find the minimal surface containing a given real analytic curve with tangent planes prescribed along the curve. We consider the generalization of this problem to non-minimal constant mean curvature (CMC) surfaces, and show that it can be solved via the loop group formulation for suc…

Multidimensional scaling (MDS) is a class of projective algorithms traditionally used in Euclidean space to produce two- or three-dimensional visualizations of datasets of multidimensional points or point distances. More recently however, several authors have pointed out that for certain datasets, hyperbolic target spa…

The article generalizes Pearson correlation to Riemannian manifolds.

Quotients $Y=X/conj$ of complex surfaces by anti-holomorphic involutions $conj\: X\to X$ tend to be completely decomposable when they are simply connected, i.e., split into connected sums, $n CP^2\#m\barCP2$, if $w_2(Y)\ne0$, or into $n(S^2\times S^2)$ if $w_2(Y)=0$. If $X$ is a double branched covering over $CP^2$, th…

Based on the model of the space $Pol_3(n)$ of polygons in $R^3$ with limited number of vertex, which was proposed by Jean-Claude Hausmann and Allen Knutson, and developed by several authors: Jason Cantarella, Alexander Y. Grosberg, Robert Kusner, and Clayton Shonkwiler, we prove that there exists an isometric isotopy o…

Simplified proof of Gaussian concentration inequality using covariance.

Simple proof for special surface classification.

Estimates area and spectrum of stable minimal surfaces in Euclidean and hyperbolic spaces.

A cone projection maps complex plane arcs to conic sections with fixed focus and directrix.

An elementary proof found for the double bubble problem in a specific norm.

The study examines when mapping class groups are quasi-isometric to graphs of curves.

Let $G = \langle A,B \rangle$ be a non-elementary two generator subgroup of the isometry group of $\mathbb{H}^2$, the hyperbolic plane. If $G$ is discrete and free and geometrically finite, its quotient is a pair of pants and in prior work we produced a formula for the number of essential self intersections (ESIs) of a…

Let $Γ<\mathrm{SL}_2(\mathbb{Z})$ be a non-elementary finitely generated subgroup and let $Γ(q)$ be its congruence subgroup of level $q$ for each $q\in \mathbb{N}$. We obtain an asymptotic formula for the matrix coefficients of $L^2(Γ(q) \backslash \mathrm{SL}_2(\mathbb{R}))$ with a {\it uniform} exponential error term…