Related papers: Primeness of alternating virtual links
We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic…
Pairwise network comparison is essential for various applications, including neuroscience, disease research, and dynamic network analysis. While existing literature primarily focuses on comparing entire network structures, we address a…
In neuroscience, researchers seek to uncover the connectivity of neurons from large-scale neural recordings or imaging; often people employ graphical model selection and estimation techniques for this purpose. But, existing technologies can…
Real-world networks may exhibit detachment phenomenon determined by the cancelling of previously existing connections. We discuss a tractable extension of Yule model to account for this feature. Analytical results are derived and discussed…
This paper gives an alternate definition of the Affine Index Polynomial (called the Wriggle Polynomial) using virtual linking numbers and explores applications of this polynomial. In particular, it proves the Cosmetic Crossing Change…
The links of a physical network cannot cross, which often forces the network layout into non-optimal entangled states. Here we define a network fabric as a two-dimensional projection of a network and propose the average crossing number as a…
Finding the dominant direction of flow of information in densely interconnected regulatory or signaling networks is required in many applications in computational biology and neuroscience. This is achieved by first identifying and removing…
Effectively showing the relationships between objects in a dataset is one of the main tasks in information visualization. Typically there is a well-defined notion of distance between pairs of objects, and traditional approaches such as…
We analyze whether preferential attachment in scientific coauthorship networks is different for authors with different forms of centrality. Using a complete database for the scientific specialty of research about "steel structures," we show…
Reciprocity characterizes the information exchange between users in a network, and some empirical studies have revealed that social networks have a high proportion of reciprocal edges. Classical directed preferential attachment (PA) models,…
To any prime alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these…
We study the enumeration of alternating links and tangles, considered up to topological (flype) equivalences. A weight $n$ is given to each connected component, and in particular the limit $n\to 0$ yields information about (alternating)…
We discuss meridians and longitudes in reduced Alexander modules of classical and virtual links. When these elements are suitably defined, each link component will have many meridians, but only one longitude. Enhancing the reduced Alexander…
Multicrossings, which have previously been defined for classical knots and links, are extended to virtual knots and links. In particular, petal diagrams are shown to exist for all virtual knots.
We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes…
Online social systems are multiplex in nature as multiple links may exist between the same two users across different social networks. In this work, we introduce a framework for studying links and interactions between users beyond the…
Networks describe a range of social, biological and technical phenomena. An important property of a network is its degree correlation or assortativity, describing how nodes in the network associate based on their number of connections.…
Let $L$ be a non-split prime alternating link with $n>0$ crossings. We show that for each fixed $g$, the number of genus-$g$ Seifert surfaces for $L$ is bounded by an explicitly given polynomial in $n$. The result also holds for all…
We construct a virtual quandle for links in lens spaces $L(p,q)$, with $q=1$. This invariant has two valuable advantages over an ordinary fundamental quandle for links in lens spaces: the virtual quandle is an essential invariant and the…
In this paper, we show that two flat fully augmented links with homeomorphic complements must be equivalent as links in $\mathbb{S}^{3}$. This requires a careful analysis of how totally geodesic surfaces and cusps intersect in these link…