On the classification of rational tangles

On the one-sided crossing minimization in a bipartite graph with large degrees

<abstract><p>The skewness of a graph $ G $, $ sk(G) $, is the smallest number of edges that need to be removed from $ G $ to make it planar. The crossing number of a graph $ G $, $ cr(G) $, is the minimum number of crossings over all possible drawings of $ G $. There is minimal work concerning the relationship between skewness and crossing numbers. In this work, we first introduce an inequality relation for these two parameters, and then we construct infinitely many near-planar graphs such that the inequality is equal. In addition, we give a necessary and sufficient condition for a graph to have its skewness equal to the crossing number and characterize some special graphs with $ sk(G) = cr(G) $.</p></abstract>

The crossing number of a graph [Formula: see text], denoted by [Formula: see text], is defined to be the minimum number of crossings that arise among all its drawings in the plane. This concept has been of interest to many researchers who have studied it for many families of graphs. In this paper, we introduce the crossing number of Cartesian product of sunlet graph [Formula: see text] with path [Formula: see text]. Further, we prove that the crossing number of [Formula: see text] is equal to [Formula: see text], along with giving a conjecture for the general case. In addition, we utilize the vertex’s rotation concept in order to prove some necessary conditions for the complete bipartite graph [Formula: see text] to be optimal when it is drawn in the plane, by presenting an upper bound for the crossing number of any subgraph in it together with determining the exact number of crossings in case the vertices of subgraph have the same rotation.

Connecting colored point sets

A book with $k$ pages consists of a straight line (the spine) and $k$ half-planes (the pages), such that the boundary of each page is the spine. If a graph is drawn on a book with $k$ pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a $k$-page drawing). The $k$-page crossing number $\nu_k(G)$ of a graph $G$ is the minimum number of crossings in a $k$-page drawing of $G$. In this paper we investigate the $k$-page crossing numbers of complete graphs. We use semidefinite programming techniques to give improved lower bounds on $\nu_k(K_n)$ for various values of $k$. We also use a maximum satisfiability reformulation to obtain a computer-aided calculation of the exact value of $\nu_k(K_n)$ for several values of $k$ and $n$. Finally, we investigate the best construction known for drawing $K_n$ in $k$ pages, calculate the resulting number of crossings, and discuss this upper bound in light of the new results reported in this paper.

In phylogenetics, tree-based networks are used to model and visualize the evolutionary history of species where reticulate events such as horizontal gene transfer have occurred. Formally, a tree-based network N consists of a phylogenetic tree T (a rooted, binary, leaf-labeled tree) and so-called reticulation edges that span between edges of T. The network N is typically visualized by drawing T downward and planar and reticulation edges with one of several different styles. One aesthetic criteria is to minimize the number of crossings between tree edges and reticulation edges. This optimization problem has not yet been researched. We show that, if reticulation edges are drawn x-monotone, the problem is NP-complete, but fixed-parameter tractable in the number of reticulation edges. If, on the other hand, reticulation edges are drawn like “ears”, the crossing minimization problem can be solved in quadratic time.

The delunification process and minimal diagrams

We study a boundary labeling problem, where multiple points may connect to the same label. In this new many-to-one model, a horizontal backbone reaches out of each label into the feature-enclosing rectangle. Feature points that need to be connected to this label are linked via vertical line segments to the backbone. We present dynamic programming algorithms for minimizing the total number of label occurrences and for minimizing the total leader length of crossing-free backbone labelings. When crossings are allowed, we aim at obtaining solutions with the minimum number of crossings. This can be achieved efficiently in the case of fixed label order; however, in the case of flexible label order we show that minimizing the number of leader crossings is NP-hard.

Computing Kauffman bracket grows exponentially with the number of crossings in the knot diagram. In this paper we illustrate how Kauffman bracket for rational tangles and rational knots can be computed so that it involves a low number of terms. Kauffman bracket and Jones polynomial are known to have connections with statistical mechanics, quantum theory and quantum field theory.

In this survey paper we sketch new combinatorial proofs of the classiflcation of rational tangles and of unoriented and oriented rational knots, using the classiflcation of alternating knots and the calculus of continued frac- tions. We continue with the classiflcation of achiral and strongly invertible rational links, and we conclude with a description of the relationships among tangles, rational knots and DNA recombination.

Unexpected behaviour of crossing sequences

Let cr(Kn) be the minimum number of crossings over all rectilinear drawings of the complete graph on n vertices in the plane. In this paper we prove that cr(Kn) < 0:380473 n + ( n 3 ); improving thus on the previous best known upper bound. This is done by obtaining new rectilinear drawings of Kn for small values of n, and then using known constructions to obtain arbitrarily large good drawings from smaller ones. The \small sets were found using a simple heuristic detailed in this paper.

We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$, for $i\in\{0,\ldots, n-1\}$. The problem of realizing $A$ by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an angle graph. In 2D, we characterize sequences $A$ for which every generic polygon $P\subset \mathbb{R}^2$ realizing $A$ has at least $c$ crossings, for every $c\in \mathbb{N}$, and describe an efficient algorithm that constructs, for a given sequence $A$, a generic polygon $P\subset \mathbb{R}^2$ that realizes $A$ with the minimum number of crossings. In 3D, we describe an efficient algorithm that tests whether a given sequence $A$ can be realized by a (not necessarily generic) polygon $P\subset \mathbb{R}^3$, and for every realizable sequence the algorithm finds a realization.

A conjecture of Shumakovitch states that every nontrivial knot has 2-torsion in its Khovanov homology.We show that if a knot K has no 2-torsion in its Khovanov homology, then the rank of its reduced Khovanov homology is minimal among all knots obtainable from K by a proper rational tangle replacement.It follows, for example, that unknotting number 1 knots have 2-torsion in their Khovanov homology. 57K18Shumakovitch conjectured that every nontrivial knot has an element of order 2 in its Khovanov homology [10, Conjecture 1].The conjecture has been verified for some infinite families of knots (see, for example, [2; 9; 10]) and has withstood large computational searches.In this note, we provide topological evidence for the conjecture, and we verify the conjecture for a large class of knots that include all unknotting number 1 knots.

Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such arcs which are disjoint from crossings. In the paper, we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.