Related papers: The Combinatorics of Alternating Tangles: from the…
New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used…
This paper is a short introduction to the theory of tangles, both in graphs and general connectivity systems. An emphasis is put on the correspondence between tangles of order k and k-connected components. In particular, we prove that there…
In recent papers we have studied refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows. The present paper is a first step towards extending these considerations to alternating sign matrices…
The preceding paper constructed tangle machines as diagrammatic models, and illustrated their utility with a number of examples. The information content of a tangle machine is contained in characteristic quantities associated to equivalence…
Comparative analyses of graph structured datasets underly diverse problems. Examples of these problems include identification of conserved functional components (biochemical interactions) across species, structural similarity of large…
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 consider a natural model of random knotting- choose a knot diagram at random from the finite set of diagrams with n crossings. We tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type, allowing us to…
A tangled-diagram over $[n]=\{1,...,n\}$ is a graph of degree less than two whose vertices $1,...,n$ are arranged in a horizontal line and whose arcs are drawn in the upper halfplane with a particular notion of crossings and nestings.…
Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured…
We study fundamental graph problems such as graph connectivity, minimum spanning forest (MSF), and approximate maximum (weight) matching in a distributed setting. In particular, we focus on the Adaptive Massively Parallel Computation (AMPC)…
Tanglegrams are a special class of graphs appearing in applications concerning cospeciation and coevolution in biology and computer science. They are formed by identifying the leaves of two rooted binary trees. We give an explicit formula…
This paper provides the complete table of prime knot projections with their mirror images, without redundancy, up to eight double points systematically thorough a finite procedure by flypes. In this paper, we show how to tabulate the knot…
In this article, we give a list of minimal grid diagrams of the 12 crossing prime alternating knots. This is a continuation of the work in https://doi.org/10.1142/S0218216520500765
A closed plane meander of order n is a closed self-avoiding loop intersecting an infinite line 2n times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm, based on…
Two types of connected chord diagrams with chord endpoints lying in a collection of ordered and oriented real segments are considered here: the real segments may contain additional bivalent vertices in one model but not in the other. In the…
This document is an exposition of an assortment of open problems arising from the exact enumeration of (perfect) matchings of finite graphs. Roughly half have been solved at the time of this writing; see the document "Twenty Open Problems…
We review the main tools which allow for the statistical characterization of weighted networks. We then present two case studies, the airline connection network and the scientific collaboration network, which are representative of critical…
We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an…
We give a sufficient condition for an almost alternating link diagram to represent a non-splittable link. The main theorem gives us a way to see if a given almost alternating link diagram represents a splittable link without increasing…
Chord diagrams and related enlacement graphs of alternating knots are enhanced to obtain complete invariant graphs including chirality detection. Moreover, the equivalence by common enlacement graph is specified and the neighborhood graph…