Related papers: Converting virtual link diagrams to normal ones
Both classical and virtual knots arise as formal Gauss diagrams modulo some abstract moves corresponding to Reidemeister moves. If we forget about both over/under crossings structure and writhe numbers of knots modulo the same Reidemeister…
A procedure of going from Heegaard diagrams to framed link diagrams is explained in this note.
The present paper is an introduction to a combinatorial theory arising as a natural generalisation of classical and virtual knot theory. There is a way to encode links by a class of `realisable' graphs. When passing to generic graphs with…
The forbidden moves can be combined with Gauss diagram Reidemeister moves to obtain move sequences with which we may change any Gauss diagram (and hence any virtual knot) into any other, including in particular the unknotted diagram
Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility,…
Twisted links are a generalization of classical links and correspond to stably equivalence classes of links in thickened surfaces. In this paper we introduce twisted intersection colorings of a diagram and construct two invariants of a…
A virtual string can be defined as an equivalence class of planar diagrams under certain kinds of diagrammatic moves. Virtual strings are related to virtual knots in that a simple operation on a virtual knot diagram produces a diagram for a…
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…
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…
We introduce a new very large family of transformations of rectangular diagrams of links that preserve the isotopy class of the link. We provide an example when two diagrams of the same complexity are related by such a transformation and…
The Reidemeister theorem states that any link in $3$-space can be encoded by a diagram (a suitably decorated projection) on a plane, and provides a finite set of combinatorial moves relating two diagrams of the same link up to isotopy. In…
Rectangular diagrams of links are link diagrams in the plane ${\mathbb R}^2$ such that they are composed of vertical line segments and horizontal line segments and vertical segments go over horizontal segments at all crossings. P. R.…
Jin and Lee proved the following: Suppose that $D_1, \dots, D_n$ are link diagrams. Given a link $L$ which is partitioned into sublinks $L_1, \dots, L_n$ admitting diagrams $D_1, \dots, D_n$ respectively, there is a diagram $D$ of $L$ whose…
We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a…
Recently N.A'Campo suggested a construction of a link from a generic immersion of a curve into a 2-disk. It is tightly related to the singularity theory. In this paper, we give a simple procedure to draw a diagram of the link from a picture…
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…
In this paper, we establish that the arc shift operation on a $n$-component virtual link diagram acts as an unknotting operation when the virtual link is $n$-homogeneous proper, aiding in the classification of \( n \)-component virtual…
We assign a new polynomial to any checkerboard-colorable 4-valent virtual graph in terms of its Euler circuit expansion. This provides a new combinatorial formulation of the Kauffman-Jones polynomial for checkerboard-colorable virtual…
We prove that a virtual link diagrams satisfying two conditions on the Khovanov homology is minimal, that is, there is no virtual diagram representing the same link with smaller number of crossings. This approach works for both classical…
Numerous studies show that most known real-world complex networks share similar properties in their connectivity and degree distribution. They are called small worlds. This article gives a method to turn random graphs into Small World…