Related papers: The monoid structure of singular twisted virtual b…

Given a representation $\varphi \colon B_n \to G_n$ of the braid group $B_n$, $n \geq 2$ into a group $G_n$, we are considering the problem of whether it is possible to extend this representation to a representation $\Phi \colon SM_n \to…

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 consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the groups of flat virtual braids and virtual Gauss braids and study some of their properties, in…

We show that a certain linear representation of the singular braid monoid on three strands is faithful. Furthermore we will give a second - group theoretically motivated - solution to the word problem in this monoid.

We study the kernels of representations of mapping class groups of surfaces on twisted homologies of configuration spaces. We relate them with the kernel of a natural twisted intersection pairing: if the latter kernel is trivial then the…

Braid groups and mapping class groups have many features in common. Similarly to the notion of inverse braid monoid inverse mapping class monoid is defined. It concerns surfaces with punctures, but among given $n$ punctures several can be…

We prove that the vector bundles at the core of the Knizhnik-Zamolodchikov and quantum constructions of braid groups representations are topologically trivial bundles. We provide partial generalizations of this result to generalized braid…

In~\cite{Ma} Manturov studied groups $G_{n}^{k}$ for fixed integers $n$ and $k$ such that $k<n$. In particular, $G_{n}^{2}$ is isomorphic to the group of free braids of $n$-stands. In~\cite{KiMa} Manturov and the author studied an invariant…

The main result of this article is that any braided (resp. annular, planar) diagram group $D$ splits as a short exact sequence $1 \to R \to D \to S \to 1$ where $R$ is a subgroup of some right-angled Artin group and $S$ a subgroup of…

Motivated by the notion of the multi-virtual braid group introduced by L. Kauffman and by the study of extensions of the well-known twin group T_n, n >= 2, we introduce a new group called the multi-virtual twin group M_kVT_n, where k >= 1…

In this mostly survey paper, we investigate the resonance varieties, the lower central series ranks, and the Chen ranks, as well as the residual and formality properties of several families of braid-like groups: the pure braid groups $P_n$,…

Machine learning (ML) is shaping our exploration of topological matter, whose existence is inherently tied to the geometry of quantum states or energy spectra. In non-Hermitian systems, distinctive spectral geometry can lead to topological…

In this paper we introduce the framed pure braid group on $n$ strands of an oriented surface, a topological generalisation of the pure braid group $P_n$. We give different equivalents definitions for framed pure braid groups and we study…

We study the problem of determining the isomorphism classes of the virtually cyclic subgroups of the n-string braid groups B_n(S^2) of the 2-sphere S^2. If n is odd, or if n is even and sufficiently large, we obtain the complete…

The singularities of the representation variety of $B_3$ are studied, where $B_3$ is the knot group on 3 strands. Specifically, we determine which semisimple representations are smooth points of this variety.

For $n\geq 2$, let $G_n$ be a group and let $\rho: B_n\rightarrow G_n$ be a representation of the braid group $B_n$. For a field $\mathbb{K}$ and $a,b,c\in \mathbb{K}$, Bardakov, Chbili, and Kozlovskaya extend the representation $\rho$ to a…

We prove the existence of a universal braided compact quantum group acting on a graph $\mathrm{C}^*$-algebra in the category of $\mathbb{T}$-$\mathrm{C}^*$-algebras with a twisted monoidal structure, in the spirit of the seminal work of S.…

We introduce a twisted fiber bundle construction of quantum CSS codes over group algebras \(R=\mathbb F_2[G]\), where each base generator carries a generator-dependent \(R\)-linear fiber twist satisfying a flatness condition. This…

This is an expository paper which provides a quick introduction to Boolean inverse semigroups and their type monoids, with the emphasis on techniques and insights of the theory, and also treats the connection of the type monoid…

In this paper we introduce and study the theories of pseudo links and singular links in the Solid Torus, ST. Pseudo links are links with some missing crossing information that naturally generalize the notion of knot diagrams, and that have…