Related papers: A criterion for almost alternating links to be non…

Link prediction is one of the fundamental problems in network analysis. In many applications, notably in genetics, a partially observed network may not contain any negative examples of absent edges, which creates a difficulty for many…

An alternating distance is a link invariant that measures how far away a link is from alternating. We study several alternating distances and demonstrate that there exist families of links for which the difference between certain…

We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link any of whose 2-component sublink is nonsplittable. We show that a graph…

It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower…

We give a simple example showing that a knot or link diagram that lies in the ${\mathbb{Z}}^2$ lattice is not necessarily the projection of a lattice stick knot or link in the ${\mathbb{Z}}^3$ lattice, and we give a necessary and sufficient…

An edge of a quasi $k$-connected graph is said to be quasi $k$-contractible if the contraction of the edge results in a quasi $k$-connected graph. We show that every 5-connected graph contains a quasi 5-contractible edge. Furthermore, we…

A realization of a virtual link diagram is obtained by choosing over/under markings for each virtual crossing. Any realization can also be obtained from some representation of the virtual link. (A representation of a virtual link is a link…

We prove several criteria for quasi-isometry between non-locally-finite graphs and their structure trees. Results of M\"oller in \cite{moeller92ends2} for locally finite and transitive graphs are generalized. We also give a criterion which…

As an extension of positive and almost positive diagrams and links, we study two classes of links we call successively almost positive and weakly successively almost positive links. We prove various properties of polynomial invariants and…

For any link in the 3-sphere, there is a natural lower bound for the unlinking number in terms of the classical signature. We prove that if this lower bound is sharp for a special alternating link $L$, then the unlinking number of $L$ is…

Here in particular, we give a characterization of Quasi-line Graphs in terms of forbidden induced subgraphs. In general, we prove a necessary and sufficient condition for a graph to be a union of two cliques.

We develop the theory of sparse multiplex networks with partially overlapping links based on their local tree-likeness. This theory enables us to find the giant mutually connected component in a two-layer multiplex network with arbitrary…

Separability problem, to decide whether a given state is entangled or not, is a fundamental problem in quantum information theory. We propose a powerful and computationally simple separability criterion, which allows us to detect the…

Using computer calculations and working with representatives of pretzel tangles we established general adequacy criteria for different classes of knots and links. Based on adequate graphs obtained from all Kauffman states of an alternating…

A long standing open conjecture states that if a link $\mathcal{K}$ is alternating, then its ropelength $L(\mathcal{K})$ is at least of the order $O(Cr(\mathcal{K}))$. A recent result shows that the maximum braid index of a link bounds the…

For a link with zero determinants, a Z-coloring is defined as a generalization of Fox coloring. We call a link having a diagram which admits a non-trivial Z-coloring a Z-colorable link. The minimal coloring number of a Z-colorable link is…

The ribbonlength of a link is a geometric invariant defined as the infimum of the ratio of the length to the width of a folded ribbon realization of the link. In this paper, we prove that if an alternating link admits an alternating diagram…

Given a class of objects, a pattern theorem is a powerful result describing their structure. We show that alternating knots exhibit a pattern theorem, and use this result to prove a long-standing conjecture that alternating knots grow rare.…

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…

Letters $x$ and $y$ alternate in a word $w$ if after deleting in $w$ all letters but the copies of $x$ and $y$ we either obtain a word $xyxy\cdots$ (of even or odd length) or a word $yxyx\cdots$ (of even or odd length). A graph $G=(V,E)$ is…