Random walks and polygons are used to model polymers. In this paper we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. W…
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48 results for “polygonal links”
Study symmetries in smoothed polygonal links.
problem Computing Khovanov homology and link invariants.
method Cube of resolutions with combinatorial structure.
result New group-theoretic invariants of links.
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 -metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexe…
The paper provides a converse to linking theorems for graphs in 3-space and higher dimensions.
problem Linking properties of graphs in 3-space and higher dimensions.
method Proves a converse to specific linking theorems for graphs in 3-space and higher dimensions.
result Proves a higher-dimensional analogue of a converse to a lemma by Segal-Spież.
We describe the construction of an multi-module in terms of counts of holomorphic polygons in a series Heegaard multi-diagrams. We show that this is quasi-isomorphic to the type-A bordered-sutured invariant of a link complement with a view to calculating, in the sequel, these invariants in terms of…
We present new computations of approximately length-minimizing polygons with fixed thickness. These curves model the centerlines of "tight" knotted tubes with minimal length and fixed circular cross-section. Our curves approximately minimize the ropelength (or quotient of length and thickness) for polygons in their kno…
The paper studies right-angled links on higher genus surfaces.
problem Classifying and understanding right-angled links on surfaces of higher genus.
method Defining and proving equivalence of properties for RGCR links, using diagram restrictions and polygonal checkerboard surfaces.
result Classification of RGCR links and bounds on their number for a given genus.
The study establishes conditions for groups acting on polygonal complexes to contain virtually free subgroups.
problem Conditions for groups acting on polygonal complexes to contain virtually free subgroups.
method Analysis of links of polygonal complexes and conditions on their structure.
result Groups acting on polygonal complexes with certain link conditions contain virtually free subgroups.
Proof of Knot Entropy Conjecture for tube lattice polygons.
problem Proving exponential growth rate of knot polygons equals unknot polygons.
method Upper and lower bounds on polygon counts, braid insertions, and pattern theorems.
result Established the Knot Entropy Conjecture for tube lattice polygons.
Paper constructs motifs from planar tilings for DP weaves and polycatenanes.
problem Creating complex entangled structures from periodic tilings.
method Combinatorial methodology using polygonal link transformations.
result Predicting the type of motif from a given tiling and polygonal link method.
Dancing polygons and rolling balls linked via a special geometric distribution.
problem Understanding the geometric and mechanical relationship between dancing polygons and rolling balls.
method Mapping dancing polygons to trajectories of a rolling ball on a 3D surface, both described by a specific geometric distribution.
result Non-degenerate dancing pairs of polygons exist for all and correspond to rolling ball trajectories.
Given a link in the three-sphere, Ozsváth and Szabó showed that there is a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched double cover. The aim of this paper is to explicitly calculate this spectral sequence in terms of bordered Floer homolo…
Utilizing both twisting and writhing, we construct integral tangles with few sticks, leading to an efficient method for constructing polygonal 2-bridge links. Let L be a two bridge link with crossing number c, stick number s, and n tangles. It is shown that s is less than or equal to 2/3 c + 2n+3 . We also show that if…
Lecture notes introduce Abelian differentials and their flat surfaces, focusing on families and Teichmüller dynamics.
problem Study of Abelian differentials and their geometric properties.
method Associate flat surfaces to Abelian differentials and analyze their families under action.
result Properties of orbit of Abelian differentials under Teichmüller dynamics.
Steinhaus conjectured that every closed oriented -curve has a pair of anti-parallel tangents. Porter disproved the conjecture by showing that there exist curves with no anti-parallel tangents. Colin Adams rised the question of whether there exists a nontrivial knot in which has no parallel or antiparallel t…
Rectangular mosaics extend virtual knot studies to larger polygons.
problem Studying virtual knots using mosaic techniques.
method Introduced rectangular mosaics, modified mosaic moves, and provided invariants.
result Developed algorithms for computing virtual knot invariants.
NT probability measures knotting in 3D arc systems.
problem Measuring knotting in 3D arc systems.
method Transforming polygonal arcs into unique diagrams, generalizing NT probability.
result Properties of NT probability for 3D arc systems are shown.
Study optimizes submatrices in 2D spaces, linking to polygon geometry.
problem Optimizing submatrices in -dimensional linear subspaces.
method Optimization problem for isoperimetric polygons in Euclidean spaces.
result New geometrical perspective on a linear subspace problem.
Non-trivialization probability of arc system in 3D space
problem Defining and generalizing the knotting probability of an arc diagram in 3D space
method Transforming polygonal arcs in 3D space into unique arc diagrams
result Introducing and generalizing the Non-Trivialization probability (NT probability) for arc systems in 3D space
Defines Vassiliev complexity measures for open and closed curves in 3D space.
problem Measuring complexity of curves in 3D space.
method Using enhanced Jones polynomial coefficients and Gauss code diagrams.
result Second Vassiliev measure converges to knot invariants as curve ends coincide.
A single-vertex origami is a piece of paper with straight-line rays called creases emanating from a fold vertex placed in its interior or on its boundary. The Single-Vertex Origami Flattening problem asks whether it is always possible to reconfigure the creased paper from any configuration compatible with the metric, t…
We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc unknotting problem} is in {\bf NP}. We also consider the problem, {\sc unknotting probl…
We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a -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…
New methods classify convex lattice polygons for affine dimers.
problem Not all convex lattice polygons are characteristic polygons of affine dimers.
method General constructions and algorithm for finding affine dimers with prescribed polygons.
result All lattice triangles, generalised parallelograms, and polygons of genus at most two admit an affine dimer.
Study on entanglement complexity of confined ring polymers in lattice tubes.
problem Understanding the entanglement complexity of confined ring polymers in lattice tubes.
method Applied knot theory to extend and prove results about the complexity of 2SAPs.
result Proved that all but exponentially few size m 2SAPs have F complexity that grows at least linearly in m as m approaches infinity.
The pentagram map's limit point is related to infinitesimal perturbations of polygons.
problem Understanding the limit point of the pentagram map and its relation to polygon perturbations.
method Interpreting Glick's operator as the infinitesimal monodromy of a polygon.
result Glick's operator measures the extent to which a perturbed polygon does not close up.
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.
problem Inverse Steklov problem on convex polygons.
method Analysis of Steklov eigenvalues and isoperimetric bounds.
result For almost all convex polygonal domains, there exist at most finitely many non-congruent domains with the same Steklov spectrum.
Study of polygon spaces, characterizing critical points of area function.
problem Characterizing critical points of area function in polygon spaces.
method Geometric characterization of critical points and calculation of Morse indices.
result Generalization of isoperimetric theorems for polygons in the plane.
New property: polygons have a fixed dimension regardless of ambient space dimensions.
problem Understanding the dimension of polygon moduli spaces.
method Generalizing the square bending example to polygons of arbitrary edge lengths.
result There are only finitely many moduli spaces of polygons with given edge lengths, even as ambient dimension increases.
The paper classifies vertices in planar polygons formed by convex domains.
problem Classifying vertices in planar polygons formed by convex domains.
method Analyzing polygons formed by homothets and translates of a convex domain.
result The number of singular boundary points in a -polygon is between and for a strictly convex domain with singular boundary points.
The pentagram map takes a planar polygon to a polygon whose vertices are the intersection points of consecutive shortest diagonals of . 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…
In this paper, we discuss centroaffine geometry of polygons in -space. For a polygon that is locally convex with respect to an origin together with a transversal vector field , we define the centroaffine dual pair similarly to [6]. We prove that vertices of 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…
Optimal Reeb graphs identified for polygon decomposition.
problem Investigating the topological structure of planar polygon decomposition.
method Using oriented Reeb graphs with a marked vertex for height functions.
result Described all possible optimal Reeb graphs for specific polygon configurations.
Study on Poncelet polygons' centers and circumcenters in various geometries.
problem Understanding Poncelet polygons' geometric centers in different geometries.
method Analyzing the Circumcenter of Mass and Center of Mass of Poncelet polygons, proving Dan Reznik's invariants, and exploring spherical geometry.
result Proof of Dan Reznik's invariants for billiard trajectories and insights into Poncelet polygons' centers in spherical geometry.
The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.
problem Finding the minimum perimeter for polygons with a fixed area in hyperbolic geometry.
method Proving analogues of the discrete isoperimetric inequality for cyclic and tangential polygons in hyperbolic geometry, considering both single and multiple polygons.
result Established two versions of the isoperimetric inequality for multiple polygons in hyperbolic geometry with certain area or perimeter restrictions.
New formula for spherical polygon area via prequantization.
problem Traditional area formula for spherical polygons requires measuring angles.
method Uses prequantization to create a new formula that doesn't require angle measurement.
result New formula applicable to a wider range of degenerate curves and polygons.
The map S transforms polygon sides, and almost no convex polygons remain convex.
problem Investigating whether convex polygons remain convex under the map S.
method Analyzing the dynamics of the map S and proving properties of the set of polygons that remain convex.
result The set of polygons that remain convex under iterations of S has measure zero and is an algebraic subvariety of codimension two.
In this article we investigate a family of nonlinear evolutions of polygons in the plane called the -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.
problem Investigating curvature and circumradius constraints for convex polygons in 2-space forms.
method Defining curvature at each vertex and proving a Blaschke-type theorem.
result The circumradius of a convex polygon satisfies a specific inequality related to its vertex curvatures.
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.
problem Understanding dual foliations of polygon spaces guided by area and perimeter.
method Investigated topology of leaves, determined homology groups, and extended isoperimetric duality.
result Homology groups and homotopy types of polygon spaces are determined.
Maximal distortion between geodesic and Euclidean diameters in polygonal domains is studied.
problem Maximal ratio of geodesic to Euclidean diameters in polygonal domains with holes.
method Analyzes convex polygons with holes, using geometric triangulations as a comparison.
result The supremum of the ratio is between and for convex polygons.
Characterizes polygonal surfaces in pseudo-hyperbolic spaces.
problem Understanding polygonal surfaces in pseudo-hyperbolic spaces.
method Characterizes polygonal surfaces by total curvature finiteness and asymptotic flatness, using comparison of ideal boundaries.
result Polygonal surfaces have parabolic type and polynomial quartic differential.
Solitons are special polygon midpoints under affine transformations.
problem Characterizing polygons whose midpoints under affine transformations form a new polygon.
method Analyzing midpoints polygons and their relationship to affine transformations and differential equations.
result A large class of polygons are on an orbit of a one-parameter subgroup of the affine group, and these curves are solutions to a specific differential equation.
Short proof for ideal polygons with near optimal orthogeodesic decomposition.
problem Decomposing ideal polygons into orthogeodesics.
method Short proof with orthogeodesic decomposition of length at most .
result Optimal orthogeodesic decomposition of ideal polygons with length .
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…