arXiv research

The paper proves rigidity of bordered polyhedral surfaces using variational principles.

The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.

A hyperbolic polygon is defined to be cyclic, horocyclic, or equidistant if its vertices lie on a metric circle, horocycle, or a component of the equidistant locus to a hyperbolic geodesic, respectively. Convex such $n$-gons are parametrized by the subspaces of $(0,\infty)^n$ that contain their side length collections,…

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, $P$ can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or…

The oriented area function $A$ is (generically) a Morse function on the space of planar configurations of a polygonal linkage. We are lucky to have an easy description of its critical points as cyclic polygons and a simple formula for the Morse index of a critical point. However, for planar polygons, the function $A$ i…

We show that an appropriate generalization of the oriented area function is a perfect Morse function on the space of three-dimensional configurations of an equilateral polygonal linkage with odd number of edges. Therefore cyclic equilateral polygons (which appear as Morse points) are interpreted as independent generato…

Paper finds optimal shapes for minimizing average lengths of billiard trajectories in specific polygons.

A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, …

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 geometricity. Wilton and the author defined the polygonality of w which also guarantees the…

We borrow a classical construction from the study of rational billiards in dynamical systems known as the "unfolding construction" and show that it can be used to study the automorphism group of a Platonic surface. More precisely, the monodromy group, or deck group in this case, associated to the cover of a regular pol…

Constructs combinatorial 2D topological field theories from cyclic A-infinity algebras.

Fixed angles of convex polygons lead to combinatorially rich polytopes.

We show that the SL(2,C)-character variety of the (-2,3,n) pretzel knot consists of two (respectively three) algebraic curves when 3 does not divide n (respectively 3 divides n) and give an explicit calculation of the Culler-Shalen seminorms of these curves. Using this calculation, we describe the fundamental polygon a…

Simplified presentation of symplectic fillings of lens spaces.

Let $K$ be a knot type for which the quadratic term of the Conway polynomial is nontrivial, and let $γ: \mathbb{R}\to \mathbb{R}^3$ be an analytic $\mathbb{Z}$-periodic function with non-vanishing derivative which parameterizes a knot of type $K$ in space. We prove that there exists a sequence of numbers $0\leq t_1 < t…

Cyclohedra are a well-known infinite familiy of finite-dimensional polytopes that can be constructed from centrally symmetric triangulations of even-sided polygons. In this article we introduce an infinite-dimensional analogue and prove that the group of symmetries of our construction is a semidirect product of a degre…

We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a $(2,n)$-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connectio…

Principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in combinatorial sense). We express rational local formulas for all…

New methods classify convex lattice polygons for affine dimers.

The pentagram map's limit point is related to infinitesimal perturbations of polygons.

General area-preserving motion of polygonal curves is formulated as a system of ODEs. Solution polygonal curves belong to a prescribed polygonal class, which is similar to the admissible class used in the crystalline curvature flow. The ODEs are discretized implicitly in time keeping a given constant area speed while s…

Study finds finitely many non-congruent polygonal domains with same Steklov spectrum.

We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we establish its validity for a large class of singularities: some rational and minimally e…

New property: polygons have a fixed dimension regardless of ambient space dimensions.

The paper classifies vertices in planar polygons formed by convex domains.

In this paper, we discuss centroaffine geometry of polygons in $3$-space. For a polygon $X$ that is locally convex with respect to an origin together with a transversal vector field $U$, we define the centroaffine dual pair $(Y,V)$ similarly to [6]. We prove that vertices of $(X,U)$ correspond to flattening points for …

We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Moebius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a ``nice'' way, and th…

The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$. This map is known to interact nicely with Poncelet polygons, i.e. polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of R. Sc…

Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic $l_1$-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexe…

Optimal Reeb graphs identified for polygon decomposition.

Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-form for the Chern class of polygon bundle. As a result an interesting combinatorics and arithmetics jumps right out of a jukebox. The calculation gives very simple rational combinatorial characteristics (we call it "curvature") o…

Study on Poncelet polygons' centers and circumcenters in various geometries.

New formula for spherical polygon area via prequantization.

The map S transforms polygon sides, and almost no convex polygons remain convex.

In this article we investigate a family of nonlinear evolutions of polygons in the plane called the $β$-polygon flow and obtain some results analogous to results for the smooth curve shortening flow: (1) any planar polygon shrinks to a point and (2) a regular polygon with five or more vertices is asymptotically stable …

The study proves a discrete Blaschke theorem for convex polygons in 2-dimensional space forms.

We study polygon spaces arising from planar configurations of necklaces with some of the beads fixed and some of the beads sliding freely. These spaces include configuration spaces of flexible polygons and some other natural polygon spaces. We characterise critical points of the oriented area function in geometric term…

Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and (-P). Then the polygon U, as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of con…

Investigates dual foliations of polygon spaces based on area and perimeter.

Characterizes polygonal surfaces in pseudo-hyperbolic spaces.

Maximal distortion between geodesic and Euclidean diameters in polygonal domains is studied.

Solitons are special polygon midpoints under affine transformations.

Short proof for ideal polygons with near optimal orthogeodesic decomposition.

We study polygonal analogues of several moving boundary problems and their time discretization which preserves the constant area speed property. We establish various polygonal analogues of geometric formulas for moving boundaries and make use of the geometric formulas for our numerical scheme and its analysis of genera…

In this paper, we study the symplectic volume of the moduli space of polygons by using Witten's formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed side-lengths. The main result of our is that among all the Spherical and Euclidean polygons with fixed perimeter the regular one…

The study connects polygon areas and projective structures in 3D space.

Fast algorithm samples confined polygons efficiently.

Persistent homology has emerged as a novel tool for data analysis in the past two decades. However, there are still very few shapes or even manifolds whose persistent homology barcodes (say of the Vietoris-Rips complex) are fully known. Towards this direction, let $P_n$ be the boundary of a regular polygon in the plane…