Related papers: Virtual quandle for links in lens spaces
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…
Virtual quandles with two operations are discussed in the article. Certain knot invariant is constructed and used to distinguish two long virtual knots.
We enhance the quandle counting invariants of oriented classical and virtual knots and links using a construction similar to quandle modules but inspired by symplectic quandle operations rather than Alexander quandle operations. Given a…
This paper studies an algebraic invariant of virtual knots called the biquandle. The biquandle generalizes the fundamental group and the quandle of virtual knots. The approach taken in this paper to the biquandle emphasizes understanding…
We introduce a quandle invariant of classical and virtual links, denoted $Q_{tc} (L)$. This quandle has the property that $Q_{tc} (L) \cong Q_{tc} (L')$ if and only if the components of $L$ and $L'$ can be indexed in such a way that $L=K_1…
Biquandles are generalizations of quandles. As well as quandles, biquandles give us many invariants for oriented classical/virtual/surface links. Some invariants derived from biquandles are known to be stronger than those from quandles for…
We introduce a notion of topological quandle. Given a topological quandle $Q$ we associate to every classical link $L$ in $\R ^3$ an invariant $J_Q(L)$ which is a topological space (defined up to a homeomorphism). The space $J_Q(L)$ can be…
We explain how the medial quandle of a classical or virtual link can be built from the peripheral structure of the reduced Alexander module.
An important geometric invariant of links in lens spaces is the lift in the 3-sphere of a link $L$ in $L(p,q)$, that is the counterimage $\widetilde L$ of $L$ under the universal covering of $L(p,q)$. If lens spaces are defined as a lens…
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…
Maps from links in thickened surfaces to flat-virtual links help to construct invariants of links using invariants of flat-virtual links. This work is dedicated to investigation of equivalence and invariants of flat-virtual diagrams…
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…
In this paper we define invariants for primitive Legendrian knots in lens spaces L(p,q) for q not equal to 1. The main invariant is a differential graded algebra which is computed from a labeled Lagrangian projection of the pair (L(p,q),…
We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.
We define new invariants of knots by means of quandle colorings and longitudinal information. These invariants can be applied to a tangle embedding problem and recognizing non-classical virtual knots.
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 introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra U_q[gl(2|1)], will be referred to as the Links-Gould invariant. We find…
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…
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 $F \times I$…
We describe a presentation for the augmented fundamental rack of a link in the lens space $L(p,1)$. Using this presentation, the (enhanced) counting rack invariants that have been defined for the classical links are applied to the links in…