Related papers: Knot Probabilities in Random Diagrams

Algorithm of construction of all knots, links with given number of crosses on diagram of knot, link is offered. This algorithm is based on simple proposition, that there is a representation of knot (link) as closure of braid with n threads…

Random graphs with a given degree sequence are often constructed using the configuration model, which yields a random multigraph. We may adjust this multigraph by a sequence of switchings, eventually yielding a simple graph. We show that,…

We compose the table of knots in the thickened torus T x I having diagrams with at most 4 crossings. The knots are constructed by the three-step process. First we list regular graphs of degree 4 with at most 4 vertices, then for each graph…

Many invariants of knots rely upon smoothing the knot at its crossings. To compute them, it is necessary to know how to count the number of connected components the knot diagram is broken into after the smoothing. In this paper, it is shown…

Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical…

Families of alternating knots (links) and tangles are studied using as building block the conway defined as the twisting of two strands. The regular representation of knots assumes the projection has the minimal number of overpassings, and…

Because of the huge number of graphs possible even with a small number of nodes, inference on network structure is known to be a challenging problem. Generating large random directed graphs with prescribed probabilities of occurrences of…

In this paper, we give the trivializing number of all minimal diagrams of positive 2-bridge knots, and study the relation between the trivializing number and the unknotting number for a part of these knots.

We study the formation of knots on a macroscopic ball-chain, which is shaken on a horizontal plate at 12 times the acceleration of gravity. We find that above a certain critical length, the knotting probability is independent of chain…

Symmetries of knots have been studied extensively, and strongly invertible knots are one of them. Lamm defined the equivariant crossing number $c_t(K)$, the minimum crossing number among all symmetric diagrams for a strongly invertible knot…

We suggest a new random model for links based on meander diagrams and graphs. We then prove that trivial links appear with vanishing probability in this model, no link $L$ is obtained with probability 1, and there is a lower bound for the…

In spatial networks vertices are arranged in some space and edges may cross. When arranging vertices in a 1-dimensional lattice edges may cross when drawn above the vertex sequence as it happens in linguistic and biological networks. Here…

We present computer simulations to examine probability distributions of gyration radius for the no-thickness closed polymers of N straight segments of equal length. We are particularly interested in the conditional distributions when the…

A veritable zoo of different knots is seen in the ensemble of looped polymer chains, whether created computationally or observed in vitro. At short loop lengths, the spectrum of knots is dominated by the trivial knot (unknot). The…

Using unknotting number, we introduce a link diagram invariant of Hass and Nowik type, which changes at most by 2 under a Reidemeister move. As an application, we show that a certain infinite sequence of diagrams of the trivial…

We introduce an unknotting-type number of knot projections that gives an upper bound of the crosscap number of knots. We determine the set of knot projections with the unknotting-type number at most two, and this result implies classical…

We study connected graphs with a fixed degree sequence, in the sparse setting where the number of edges grows linearly in the number of vertices. Using the relation to the configuration model, we identify the number of such connected graphs…

An analysis of extensive simulations of interacting self-avoiding polygons on cubic lattice shows that the frequencies of different knots realized in a random, collapsed polymer ring decrease as a negative power of the ranking order, and…

A knot is a closed loop in space without self-intersection. Two knots are equivalent if there is a self homeomorphism of space bringing one onto the other. An arc presentation is an embedding of a knot in the union of finitely many half…

We study graphs that are formed by independently-positioned needles (i.e., line segments) in the unit square. To mathematically characterize the graph structure, we derive the probability that two line segments intersect and determine…