Related papers: Knot probabilities in equilateral random polygons

We develop a general method for constructing random manifolds and submanifolds in arbitrary dimensions. The method is based on associating colors to the vertices of a triangulated manifold, as in recent work for curves in 3-dimensional…

The interplay of geometrical and topological entanglement in semiflexible knotted polymer rings confined inside a spherical cavity is investigated using advanced numerical methods. By using stringent and robust algorithms for locating…

We apply a probabilistic approach to study the computational complexity of analog computers which solve linear programming problems. We analyze numerically various ensembles of linear programming problems and obtain, for each of these…

An equilateral stick number $s_{=}(K)$ of a knot $K$ is defined to be the minimal number of sticks required to construct a polygonal knot of $K$ which consists of equal length sticks. Rawdon and Scharein [12] found upper bounds for the…

We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a…

We prove that for knots, the evaluation of the Jones polynomial at the sixth root of unity, as well as the evaluation of the $Q$-polynomial at the reciprocal of the golden ratio, are uniquely determined by the oriented homeomorphism type of…

We prove that the detection rate of n-crossing alternating links by many standard link invariants decays exponentially in n, implying that they detect alternating links with probability zero. This phenomenon applies broadly, in particular…

Polygonal slap maps are piecewise affine expanding maps of the interval obtained by projecting the sides of a polygon along their normals onto the perimeter of the polygon. These maps arise in the study of polygonal billiards with…

Link equivalence up to isotopy in a 3-space is the problem that lies at the root of knot theory, and is important in 3-dimensional topology and geometry. We consider its restriction to alternating links, given by two alternating diagrams…

Since quantum spatial searches on complex networks have a strong network dependence, the question arises whether the universal perspective exists in this quantum algorithm for complex networks. Here, we uncover the universal scaling laws of…

A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by…

This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, Kaufman two-variable polynomial, and Khovanov polynomial.

Latent Gaussian models have a rich history in statistics and machine learning, with applications ranging from factor analysis to compressed sensing to time series analysis. The classical method for maximizing the likelihood of these models…

In this paper we prove that the Casson-Gordon invariants of the connected sum of two knots split when the Alexander polynomials of the knots are coprime. As one application, for any knot K, all but finitely many algebraically slice twisted…

A model of random walk on knot diagrams is used to study the Alexander polynomial and the colored Jones polynomial of knots. In this context, the inverse of the Alexander polynomial of a knot plays the role of an Ihara-Selberg zeta function…

The volume conjecture and its generalization state that the series of certain evaluations of the colored Jones polynomials of a knot would grow exponentially and its growth rate would be related to the volume of a three-manifold obtained by…

This manuscript introduces a new framework for the study of knots by exploring the neighborhood of knot embeddings in the space of simple open and closed curves in 3-space. The latter gives rise to a knotoid spectrum, which determines the…

We prove the fractal crumpled structure of collapsed unknotted polymer ring. In this state the polymer chain forms a system of densely packed folds, mutually separated in all scales. The proof is based on the numerical and analytical…

Global textures are interesting because they are promising candidates for seeds in the structure formation of the universe. The important configurations are those who will be able to collapse. The type of configurations that I will consider…

Anomalously strong finite-size effects have been observed for the mean square radius of gyration $R^2_K$ of Gaussian random polygons with a fixed knot $K$ as a function of the number $N$ of polygonal nodes. Through computer simulations with…