Narrow Results

We classify closed virtual 2-braids completely as virtual links. For the proof, we use surface bracket polynomial due to Dye and Kauffman, Kuperberg's theorem which states existence and uniqueness of a minimal realization for a virtual link, and a subgroup of the 3-braid group.

We introduce a recoupling theory for virtual braided trees. This recoupling theory can be utilized to incorporate swap gates into anyonic models of quantum computation.

There is a well-known injective homomorphism $\varphi :{{ℬ}}_n\to \mathrm{\text{Aut}}(F_n)$ from the classical braid group ${{ℬ}}_n$ into the automorphism group of the free group $F_n$, first described by Artin [Theory of Braids, Ann. Math. (2) 48(1) (1947) 101–126]. This homomorphism induces an action of ${{ℬ}}_n$ on $F_n$ that can be recovered by considering the braid group as the mapping class group of $H_n$ (an upper half plane with $n$ punctures) acting naturally on the fundamental group of $H_n$. Kauffman introduced virtual links [Virtual knot theory, European J. Combin.20 (1999) 663–691] as an extension of the classical notion of a link in ${ℝ}^3$. There is a corresponding notion of a virtual braid, and the set of virtual braids on $n$ strands forms a group ${{𝒱}{ℬ}}_n$. In this paper, we will generalize the Artin action to virtual braids. We will define a set, ${{𝒱}{𝒞}{𝒟}}_n$, of “virtual curve diagrams” and define an action of ${{𝒱}{ℬ}}_n$ on ${{𝒱}{𝒞}{𝒟}}_n$. Then, we will show that, as in Artin’s case, the action is faithful. This provides a combinatorial solution to the word problem in ${{𝒱}{ℬ}}_n$. In the papers [V. G. Bardakov, Virtual and welded links and their invariants, Siberian Electron. Math. Rep.21 (2005) 196–199; V. O. Manturov, On recognition of virtual braids, Zap. Nauch. Sem. POMI299 (2003) 267–286], an extension $\psi :{{𝒱}{ℬ}}_n\to \mathrm{\text{Aut}}(F_{n+1})$ of the Artin homomorphism was introduced, and the question of its injectivity was raised. We find that $\psi$ is not injective by exhibiting a non-trivial virtual braid in the kernel when $n=4$.

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, stability and Reidemeister moves. We show that virtual braids are in a bijective correspondence with abstract braids. Finally we demonstrate that for any abstract braid, its representative of minimal genus is unique up to compatibility and Reidemeister moves. The genus of such a representative is thus an invariant for virtual braids. We also give a complete proof of the fact that there is a bijective correspondence between virtually equivalent virtual braid diagrams and braid-Gauss diagrams.

In the present paper, a new representation of the virtual braid group ${VB}_n$ into the automorphism group of free product of the free group and free abelian group is constructed. This representation generalizes the previously constructed ones. The fact that the previously known representations are not faithful for $n\ge 4$ is verified. Using representations of ${VB}_n$, a virtual link group is defined. Also representations of the welded braid group ${WB}_n$ are constructed and the welded link group is defined.

In the paper [13], for an arbitrary virtual link $L$, three groups $G_{1,r}(L)$, $r>0$, $G_2(L)$ and $G_3(L)$ were defined. In the present paper, these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to each other is proved. Also, we prove that $G_3$ distinguishes the Kishino knot from the trivial knot. The fact that these groups have the lower central series which does not stabilize on the second term is noted. Hence, we have a possibility to study these groups using quotients by terms of the lower central series and to construct representations of these groups in rings of formal power series. It allows to construct an invariants for virtual knots.

In this paper, we discuss algebraic, combinatorial and topological properties of singular virtual braids. On the algebraic side, we state the relations between classical and singular virtual objects, in addition we discuss a Birman-like conjecture for the virtual case. On the topological and combinatorial side, we prove that there is a bijection between singular abstract braids, singular horizontal Gauss diagrams up to a certain equivalence relation, and singular virtual braids, in particular using singular horizontal Gauss diagrams we obtain a presentation of the singular pure virtual braid monoid.

Over-then-Under (OU) tangles are oriented tangles whose strands travel through all of their over crossings before any under crossings. In this paper, we discuss the idea of gliding: an algorithm by which tangle diagrams could be brought to OU form. By analyzing cases in which the algorithm converges, we obtain a braid classification result, which we also extend to virtual braids, and provide a Mathematica implementation. We discuss other instances of successful “gliding ideas” in the literature — sometimes in disguise — such as the Drinfel’d double construction, Enriquez’s work on quantization of Lie bialgebras, and Audoux and Meilhan’s classification of welded homotopy links.

In this paper we sketch our proof of a Markov Theorem for the virtual braid group using L-move techniques.