Related papers: Wirtinger Numbers for Virtual Links
We define the {\it Wirtinger number} of a link, an invariant closely related to the meridional rank. The Wirtinger number is the minimum number of generators of the fundamental group of the link complement over all meridional presentations…
A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives…
Given a virtual link diagram $D$, we define its unknotting index $U(D)$ to be minimum among $(m, n)$ tuples, where $m$ stands for the number of crossings virtualized and $n$ stands for the number of classical crossing changes, to obtain a…
The splitting number of a link is the minimal number of crossing changes between different components required to convert it into a split link. We obtain a lower bound on the splitting number in terms of the (multivariable) signature and…
The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings. All previously known estimates for…
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…
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…
The splitting number of a link is the minimum number of crossing changes between distinct components that is required to convert the link into a split link. We provide a bound on the splitting number in terms of the four-genus of related…
Using Gauss diagrams, one can define the virtual bridge number ${\rm vb}(K)$ and the welded bridge number ${\rm wb}(K),$ invariants of virtual and welded knots with ${\rm wb}(K) \leq {\rm vb}(K).$ If $K$ is a classical knot, Chernov and…
We compute lower bounds on the virtual crossing number and minimal surface genus of virtual knot diagrams from the arrow polynomial. In particular, we focus on several interesting examples.
This paper employs various computational techniques to determine the bridge numbers of both classical and virtual knots. For classical knots, there is no ambiguity of what the bridge number means. For virtual knots, there are multiple…
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 introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus.
We define the virtual bridge number $vb(K)$ and the virtual unknotting number $vu(K)$ invariants for virtual knots. For ordinary knots $K$ they are closely related to the bridge number $b(K)$ and the unknotting number $u(K)$ and we have…
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…
We extend the Wirtinger number of links, an invariant originally defined by Blair, Kjuchukova, Velazquez, and Villanueva in terms of extending initial colorings of some strands of a diagram to the entire diagram, to spatial graphs. We prove…
The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering…
In this paper, we compute the slice genus for many low-crossing virtual knots. For instance, we show that 1295 out of 92800 virtual knots with 6 or fewer crossings are slice, and that all but 248 of the rest are not slice. Key to these…
We prove the meridional rank conjecture for twisted links and arborescent links associated to bipartite trees with even weights. These links are substantial generalizations of pretzels and two-bridge links, respectively. Lower bounds on…
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…