Related papers: Virtual Genus of Satellite Links
The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot…
This paper presents evidence supporting the surprising conjecture that in the topological category the slice genus of a satellite knot $P(K)$ is bounded above by the sum of the slice genera of $K$ and $P(U)$. Our main result establishes…
Our Milky Way galaxy is host to a number of companions. These companions are gravitationally bound to the Milky Way and are stellar systems in their own right. They include a population of some 30 dwarf satellite galaxies (DSGs) and about…
Virtual knots are defined diagrammatically as a collection of figures, called virtual knot diagrams, that are considered equivalent up to finite sequences of extended Reidemeister moves. By contrast, knots in $\mathbb{R}^3$ can be defined…
Twin groups and virtual twin groups are planar analogues of braid groups and virtual braid groups, respectively. These groups play the role of braid groups in the Alexander-Markov correspondence for the theory of stable isotopy classes of…
For digital images, there is an established homotopy equivalence relation which parallels that of classical topology. Many classical homotopy equivalence invariants, such as the Euler characteristic and the homology groups, do not remain…
The virtual observatory (VO) is a collection of interoperable data archives, tools and applications that together form an environment in which original astronomical research can be carried out. The VO is opening up new ways of exploiting…
Computing layer similarities is an important way of characterizing multiplex networks because various static properties and dynamic processes depend on the relationships between layers. We provide a taxonomy and experimental evaluation of…
Properties of pairs of product conjugate connections are stated with a special view towards the integrability of the given almost product structure. We define the analogous in product geometry of the structural and the virtual tensors from…
In this paper we prove a Markov Theorem for virtual braids and for some analogs of this structure. The virtual braid group is the natural companion in the category of virtual knots, just as the Artin braid group is the natural companion to…
In this note we make use of some properties of vector fields on a manifold to give an alternate proof to [3] for the equivalence between connections and parallel transport on vector bundles over manifolds. Out of the proof will emerge a new…
Digital nets (in base $2$) are the subsets of $[0,1]^d$ that contain the expected number of points in every not-too-small dyadic box. We construct sets that contain almost the expected number of points in every such box, but which are…
A classical link in 3-space can be represented by a Gauss paragraph encoding a link diagram in a combinatorial way. A Gauss paragraph may code not a classical link diagram, but a diagram with virtual crossings. We present a criterion and a…
The connected sum of two flat virtual knots depends on the choice of diagrams and basepoints. We show that any minimal crossing diagram of a composite flat virtual knot is a connected sum diagram. We also show the crossing number of flat…
A knot K is called n-adjacent to the unknot, if K admits a projection containing n generalized crossings such that changing any m (no larger than n) of them yields a projection of the unknot. We show that a non-trivial satellite knot K is…
We address the problem of link reciprocity, the non-random presence of two mutual links between pairs of vertices. We propose a new measure of reciprocity that allows the ordering of networks according to their actual degree of correlation…
Kuperberg [Algebr. Geom. Topol. 3 (2003) 587-591] has shown that a virtual knot corresponds (up to generalized Reidemeister moves) to a unique embedding in a thichened surface of minimal genus. If a virtual knot diagram is equivalent to a…
This paper studies knots in three dimensional projective space. Our technique is to associate a virtual link to a link in projective space so that equivalent projective links go to equivalent virtual links (modulo a special flype move). We…
Many functions have been recently defined to assess the similarity among networks as tools for quantitative comparison. They stem from very different frameworks - and they are tuned for dealing with different situations. Here we show an…
Using a new tool called lassos, we establish a new correspondence between cellular link {diagrams} on closed surfaces and equivalence classes of virtual link {diagrams}. This is analogous to a well-known correspondence among the links…