Related papers: Multi-virtual braid groups

A new method for deriving universal \v{R} matrices from braid group representation is discussed. In this case, universal \v{R} operators can be defined and expressed in terms of products of braid group generators. The advantage of this…

We introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study properties. Our approach is slightly different from the one introduced in the thesis of S.~Roy and, unlike the investigations…

In this paper, we introduce twisted virtual doodles, defined as stable equivalence classes of immersed circles on closed surfaces that may be non-orientable. These objects admit planar representative diagrams, considered up to a suitable…

This article is an exposition of certain connections between the braid groups, classical homotopy groups of the 2-sphere, as well as Lie algebras attached to the descending central series of pure braid groups arising as Vassiliev invariants…

We give a survey of the theory of surface braid groups and the lower algebraic K-theory of their group rings. We recall several definitions and describe various properties of surface braid groups, such as the existence of torsion,…

We study the algebraic structures of the virtual singular braid monoid, $VSB_n$, and the virtual singular pure braid monoid, $VSP_n$. The monoid $VSB_n$ is the splittable extension of $VSP_n$ by the symmetric group $S_n$. We also construct…

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,…

In 2015 Hikami and Inoue constructed a representation of the braid group in terms of cluster algebra associated with the decomposition of the complement of the corresponding knot into ideal hyperbolic tetrahedra. This representation leads…

In this article, we give a necessary and sufficient condition for embedding a finite index subgroup of Artin's braid group into the mapping class group of a connected orientable surface.

In this article we prove theorem on Lifting for the set of virtual pure braid groups. This theorem says that if we know presentation of virtual pure braid group $VP_4$, then we can find presentation of $VP_n$ for arbitrary $n > 4$. Using…

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided…

We study combinatorial properties of virtual braid groups and we describe relations with finite type invariant theory for virtual knots and Yang-Baxter equations

Virtual links were introduced by Kauffman in 1999. We characterize the virtual link invariants that are partition functions of vertex models (as considered by de la Harpe and Jones), both in the real and in the complex case. We show that…

We investigate the resonance varieties, lower central series ranks, and Chen ranks of the pure virtual braid groups and their upper-triangular subgroups. As an application, we give a complete answer to the 1-formality question for this…

We introduce the relative Coxeter groupoid and construct intrinsic relative braid group symmetries for quantum supersymmetric pairs of type sAIII. These symmetries are constructed by establishing new intertwining properties of quasi…

We investigate braid group representations associated with unitary braided vector spaces, focusing on a conjecture that such representations should have virtually abelian images in general and finite image provided the braiding has finite…

The role of quantum groups and braid groups in the description of Standard Model particles is discussed. Some recent results on the use of the quantum group $SU_q(3)$ as a flavour symmetry are reviewed and a connection between two…

We study some examples of braided categories and quasitriangular Hopf algebras and decide which of them is pseudosymmetric, respectively pseudotriangular. We show also that there exists a universal pseudosymmetric braided category.

A near-group category is an additively semisimple category with a product such that all but one of the simple objects is invertible. We classify braided structures on near-group categories, and give explicit numerical formulas for their…

We construct a homomorphism $f$ from the braid group $B_{2n+2}$ on $2n+2$ strands to the Steinberg group associated with the Lie type $C_n$ and with integer coefficients. This homomorphism lifts the well-known symplectic representation of…