Related papers: Virtual links which are equivalent as twisted link…
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…
We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also…
Twisted knot theory, introduced by M.O. Bourgoin, is a generalization of virtual knot theory. It naturally yields the notion of a twisted braid, which is closely related to the notion of a virtual braid due to Kauffman. In this paper, we…
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative.In this paper, we introduce a method of converting a…
We apply the twisting technique that was first introduced in \cite{CK} and later generalized in \cite{QCQ} to obtain an infinite family of adequate, homogeneous or alternative links from a given adequate, homogeneous or alternative link,…
Virtual index cocycle is the 1-cochain that counts virtual crossings in the arcs of a virtual link diagram. We show how this cocycle can be used to reformulate and unify some known invariants of virtual links.
A link diagram can be considered as a $4$-valent graph embedded in the $2$-sphere and divides the sphere into complementary regions. In this paper, we show that any link has a diagram with only triangles and quadrilaterals. This extends…
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…
An oriented link projection is the image of a generic immersion of oriented circles into the 2-sphere. The circle arrangement of a link projection is a disjoint union of unoriented circles on the 2-sphere obtained by orientation-incoherent…
The virtual genus of a virtual satellite link is equal to that of its companion.
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…
We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also…
In 2002, D. Hrencecin and L.H. Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its…
A link stream is a set of possibly weighted triplets (t, u, v) modeling that u and v interacted at time t. Link streams offer an effective model for datasets containing both temporal and relational information, making their proper analysis…
Twisted knot theory introduced by M. Bourgoin is a generalization of knot theory. It leads us to the notion of twisted virtual braids. In this paper we show theorems for twisted links corresponding to the Alexander theorem and the Markov…
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 local deformation called the virtualized $\Delta$-move for virtual knots and links. We prove that the virtualized $\Delta$-move is an unknotting operation for virtual knots. Furthermore we give a necessary and sufficient…
For any virtual link, a class of new links can be defined called stacks, in which copies of the virtual link are placed on top of one another. The resulting virtual link depends only on the virtual isotopy class of the original link, and…
Cobordism of virtual string links on $n$ strands is a combinatorial generalization of link cobordism. There exists a bijection between virtual string links up to cobordisms and elements of the group $\mathbb{Z}^{n(n-1)}$. This paper also…
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…