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In this paper, using some properties of fundamental groups and covering spaces of connected polyhedra and CW-complexes, we present topological proof for some famous theorems about finitely presented groups.
We provide explicit and unified formulae for the normalized 3-cocycles on arbitrary finite abelian groups. As an application, we compute all the braided monoidal structures on linear Gr-categories.
We prove that representations of the braid groups coming from weakly group-theoretical braided fusion categories have finite images.
In this paper, we define the set of singular grid diagrams $\mathcal{SG}$ which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set…
Lawrence-Krammer representations (LK-representations for short) are linear representations of Artin-Tits groups of small type, which are of importance since they are known to be faithful when the type is spherical, or more generally when…
We give an explicit description of a fibration of the complement of the closure of a homogeneous braid, understanding how each fiber intersects every cross-section of $S^3$.
We expound the properties of ribbons in a setting which is general enough to encompass spherical Artin monoids and dual braid monoids of well-generated complex reflection groups. We generalize to our setting results on parabolic subgroups…
We introduce a monoidal category whose morphisms are finite partial orders, with chosen minimal and maximal elements as source and target respectively. After recalling the notion of presentation of a monoidal category by the means of…
Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid…
In this article, we study the irreducibility of representations of the singular braid group on $n$ strands, namely $SB_n$. Our first finding is the determination of the forms of all irreducible representations $\rho : SB_2 \to…
In this paper we introduce a strict monoidal subcategory of the category of matrices, suitable to address a higher representation theoretic analogue of radicals (non-semisimplicity) in ordinary representation theory. We show the extent to…
We describe the structure of finite Boolean inverse monoids and apply our results to the representation theory of finite inverse semigroups. We then generalize to semisimple Boolean inverse semigroups.
The node-opening technique, originally designed for constructing minimal surfaces, is adapted to construct a rich variety of new maxfaces of high genus that are embedded outside a compact set and have arbitrarily many catenoid or planar…
This article is about Artin's braid group and its role in knot theory. We set ourselves two goals: (i) to provide enough of the essential background so that our review would be accessible to graduate students, and (ii) to focus on those…
In this expositional essay, we introduce some elements of the study of groups by analysing the braid pattern on a knitted blanket. We determine that the blanket features pure braids with a minimal number of crossings. Moreover, we determine…
We generalize the Moishezon Teicher algorithm that was suggested for the computation of the braid monodromy of an almost real curve. The new algorithm suits a larger family of curves, and enables the computation of braid monodromy not only…
For every group genetic code with finite number of generating and at most with one defining relation we introduce the braid group of this genetic code. This construction includes the braid group of Euclidean plane, the braid groups of…
A combinatorial property of prositive group presentations, called completeness, is introduced, with an effective criterion for recognizing complete presentations, and an iterative method for completing an incomplete presentation. We show…
We prove a compactness theorem for metrics with Bounded Integral Curvature on a fixed closed surface $\Sigma$. As a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities, where an accumulation…
We consider exact sequences and lower central series of surface braid groups and we explain how they can prove to be useful for obtaining representations for surface braid groups. In particular, using a completely algebraic framework, we…