Related papers: On virtual link invariants
We define new notions of groups of virtual and welded knots (or links) and we study their relations with other invariants, in particular the Kauffman group of a virtual knot.
Multi-virtual knot theory was introduced in $2024$ by the first author. In this paper, we initiate the study of algebraic invariants of multi-virtual links. After determining a generating set of (oriented) multi-virtual Reidemeister moves,…
We characterize the virtual link invariants that can be described as partition function of a real-valued R-matrix, by being weakly reflection positive. Weak reflection positivity is defined in terms of joining virtual link diagrams, which…
The affine index polynomial and the $n$-writhe are invariants of virtual knots which are introduced by Kauffman and by Satoh and Taniguchi independently. They are defined by using indices assigned to each classical crossing, which we call…
Virtual knot theory, introduced by Kauffman, is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation of Etingof and Kazhdan's theory of quantization of…
A polynomial invariant of virtual links, arising from an invariant of links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur, is defined and its properties are investigated. Examples are given that the invariant can detect…
We define a multi-variable version of the Affine Index Polynomial for virtual links. This invariant reduces to the original Affine Index Polynomial in the case of virtual knots, and also generalizes the version for compatible virtual links…
We introduce new topological quantum invariants of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed via surgery on manifolds of the form $F \times I$…
In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of…
We introduce stable equivalence classes of oriented links in orientable three-manifolds that are orientation $I$-bundles over closed but not necessarily orientable surfaces. We call these twisted links, and show that they subsume the…
This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. The paper sets up…
Virtual knots, defined by Kauffman, provide a natural generalization of classical knots. Most invariants of knots extend in a natural way to give invariants of virtual knots. In this paper we study the fundamental groups of virtual knots…
A group invariant for links in thickened closed orientable surfaces is studied. Associated polynomial invariants are defined. The group detects nontriviality of a virtual link and determines its virtual genus.
For a virtual $n$-link $K$, we define a new virtual link $VD(K)$, which is invariant under virtual equivalence of $K$. The Dehn space of $VD(K)$, which we denote $DD(K)$, therefore has a homotopy type which is an invariant of $K$. We show…
In a recent paper Jones introduced a correspondence between elements of the Thompson group $F$ and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be…
In the present paper, we describe new approaches for constructing virtual knot invariants. The main background of this paper comes from formulating and bringing together the ideas of biquandle (Kauffman and Radford) the virtual quandle…
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…
A virtual link can be understood as a link in a trivial I-bundle over an orientable compact surface with genus. A twisted virtual link is a link in a trivial I-bundle over a not-necessarily orientable compact surface. A twisted virtual…
F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman's affine index polynomial and use smoothing in classical crossing of a…
In the paper we introduce a general approach how for a given virtual biquandle multi-switch $(S,V)$ on an algebraic system $X$ (from some category) and a given virtual link $L$ construct an algebraic system $X_{S,V}(L)$ (from the same…