Related papers: Models of Random Knots

The interplay of topological constraints and Coulomb interactions in static and dynamic properties of charged polymers is investigated by numerical simulations and scaling arguments. In the absence of screening, the long-range interaction…

We discuss the possibility of the existence of finite algorithms that may give distinct knot classes. In particular we present two attempts for such algorithms which seem promising, one based on knot projections on a plane, the other on…

In thermally fluctuating long linear polymeric chain in solution, the ends come from time to time into a direct contact or a close vicinity of each other. At such an instance, the chain can be regarded as a closed one and thus will form a…

We introduce and study knots and links in 2-dimensional complexes. In particular, we define linking numbers for oriented two-component links in 2-complexes and a Kauffman-type bracket polynomial for links in 2-complexes. We also discuss…

We consider random graphs sampled uniformly from a structured class of graphs, such as the class of graphs embeddable in a given surface. We sharpen and extend earlier results on pendant appearances, concerning for example numbers of…

The motivation for this work was to construct a nontrivial knot with trivial Jones polynomial. Although that open problem has not yielded, the methods are useful for other problems in the theory of knot polynomials. The subject of the…

Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via…

We construct new knot polynomials. Let $V$ be the standard solid torus in 3-space and let $pr$ be its standard projection onto an annulus. Let $M$ be the space of all smooth oriented knots in $V$ such that the restriction of $pr$ is an…

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 study random pinning and copolymer models, when the return distribution of the underlying renewal process has a polynomial tail with finite mean. We compute the asymptotic behavior of the critical curves of the models in the weak…

Knots are deeply entangled with every branch of science. One of the biggest open challenges in knot theory is to formalise a knot invariant that can unambiguously and efficiently distinguish any two knotted curves. Additionally, the…

We present a systematic classification of uncolored bonded knots with singularity number at most seven. Bonded knots provide a topological model for closed protein chains with intramolecular bridges, such as disulfide bonds. Following the…

2-dimensional knots and links are studied in the article. The notion of parity is introduced via techniques similar to the ones used by the second named author in 1-dimensional case. By using parity new invariants are constructed and known…

We study the behavior of Legendrian and transverse knots under the operation of connected sums. As a consequence we show that there exist Legendrian knots that are not distinguished by any known invariant. Moreover, we classify Legendrian…

In this work the thermodynamic properties of short polymer knots (up to 120 segments) defined on a simple cubic lattice are studied with the help of the Wang-Landau Monte Carlo algorithm. The sampling process is performed using pivot…

We initiate the study of classical knots through the homotopy class of the n-th evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its n-th evaluation map realizes the…

Knot diagrams are among the most common visual tools in topology. Computer programs now make it possible to draw, manipulate and render them digitally, which proves to be useful in knot theory teaching and research. Still, an openly…

Random tensor networks are a powerful toy model for understanding the entanglement structure of holographic quantum gravity. However, unlike holographic quantum gravity, their entanglement spectra are flat. It has therefore been argued that…

This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the…

This paper is expository and is accessible to students. We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible…