Related papers: On PM-monoids and braid PM-monoids
A special class of braids, called woven, is introduced and it is shown that every conjugation class of the braid group contains woven braids. In consequence, links can be presented as plats or closures of woven braids. Restricting on knots,…
We detail a construction of a symmetric monoidal structure, called the reduced tensor product on the 2-category of braided tensor categories $\mathbf{BTC}(\mathcal{A})$ containing a fixed symmetric fusion subcategory $\mathcal{A}$. The…
Empirical properties of generating systems for complex reflection groups and their braid groups have been observed by Orlik-Solomon and Brou\'e-Malle-Rouquier, using Shephard-Todd classification. We give a general existence result for…
Certain aspects of Street's formal theory of monads in 2-categories are extended to multimonoidal monads in symmetric strict monoidal 2-categories. Namely, any symmetric strict monoidal 2-category $\mathcal M$ admits a symmetric strict…
Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also…
We unify and generalize several approaches to constructing braid group representations from finite groups, using iterated twisted tensor products. Our results hint at a relationship between the braidings on the $G$-gaugings of a pointed…
We construct a new monoid structure for Artin groups associated with finite Coxeter systems. This monoid shares with the classical positive braid monoid a crucial algebraic property: it is a Garside monoid. The analogy with the classical…
This is the last part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans - crossed…
The concepts of twisted knot theory and singular knot theory inspire the introduction of singular twisted knot theory. This study showcases similar findings for singular twisted links, including the Alexander theorem and the Markov theorem…
We associate to every positive braid a braid monodromy group, generalizing the geometric monodromy group of an isolated plane curve singularity. If the closure of the braid is a knot, we identify the corresponding group with a framed…
We develop a notion of iterated monoidal category and show that this notion corresponds in a precise way to the notion of iterated loop space. Specifically the group completion of the nerve of such a category is an iterated loop space and…
We study categorical models for the unitless fragment of multiplicative linear logic. We find that the appropriate notion of model is a special kind of promonoidal category. Since the theory of promonoidal categories has not been developed…
In the last decade, a number of public key cryptosystems based on com- binatorial group theoretic problems in braid groups have been proposed. We survey these cryptosystems and some known attacks on them. This survey includes: Basic facts…
A framework of monomial codes is considered, which includes linear codes generated by the evaluation of certain monomials. Polar and Reed-Muller codes are the two best-known representatives of such codes and can be considered as two extreme…
The Standard Model of particle physics provides very accurate predictions of phenomena occurring at the sub-atomic level, but the reason for the choice of symmetry group and the large number of particles considered elementary, is still…
We introduce the notion of a braid group parametrized by a ring, which is defined by generators and relations and based on the geometric idea of painted braids. We show that the parametrized braid group is isomorphic to the semi-direct…
Braidoids generalize the classical braids and form a counterpart theory to the theory of planar knotoids, just as the theory of braids does for the theory of knots. In this paper, we introduce basic notions of braidoids, a closure operation…
We consider a natural generalization of braids which we call shrinking braids. We state the relations of shrinking braids and use them to define algebraically the monoid $R$. We endow a subset of $R$ with a \emph{left distributive monoid}…
The mixed braid groups are the subgroups of Artin braid groups whose elements preserve a given partition of the base points. We prove that the centralizer of any braid can be expressed in terms of semidirect and direct products of mixed…
A braided generalization of the concept of Hopf algebra (quantum group) is presented. The generalization overcomes an inherent geometrical inhomogeneity of quantum groups, in the sense of allowing completely pointless objects. All…