arXiv subjects
Publications and source records attributed to Celeste Damiani.
Following previous work on congruence subgroups and crystallographic braid groups, we study the lower central series of congruence braid groups related to the braid group $B_3$, showing in particular that corresponding quotients are almost crystallographic.
In this work, we study the relationship between congruence subgroups $B_n[m]$ and $\mathcal{N}_n(σ_1^m)$ the normal closure of $σ_1^m$, where $σ_1$ is the classical generator of $B_n$. We characterize the conditions under which $\mathcal{N}_n(σ_1^m)$ has finite index in $B_n[m]$ and provide explicit generators for these finite quotients. For the cases where the index is infinite, we show that $B_n[m]/\mathcal{N}_n(σ_1^m)$ contains a free subgroup. Additionally, we compute the Abelianisation of Coxeter braid subgroups in the finite index cases and construct new finite quotients using commutators of congruence subgroups.
Federated learning is becoming an increasingly viable and accepted strategy for building machine learning models in critical privacy-preserving scenarios such as clinical settings. Often, the data involved is not limited to clinical data but also includes additional omics features (e.g. proteomics). Consequently, data is distributed not only across hospitals but also across omics centers, which are labs capable of generating such additional features from biosamples. This scenario leads to a hybrid setting where data is scattered both in terms of samples and features. In this hybrid setting, we present an efficient reformulation of the Kernel Regularized Least Squares algorithm, introduce two variants and validate them using well-established datasets. Lastly, we discuss security measures to defend against possible attacks.
This paper is the first of a two part series devoted to describing relations between congruence and crystallographic braid groups. We recall and introduce some elements belonging to congruence braid groups and we establish some (iso)-morphisms between crystallographic braid groups and corresponding quotients of congruence braid groups.
We introduce the notion of wen knots, and prove that the set of wen knots is a proper subset of the set of extended welded knots. Furthermore we prove that the complementary subset consists of welded knots up to horizontal mirror reflections. This allow us to characterise completely extended welded knots by the parity of their number of wens, that we can always reduce to 0 or 1.
We introduce a generalisation $LH_n$ of the ordinary Hecke algebras informed by the loop braid group $LB_n$ and the extension of the Burau representation thereto. The ordinary Hecke algebra has many remarkable arithmetic and representation theoretic properties, and many applications. We show that $LH_n$ has analogues of several of these properties. In particular we %introduce consider a class of local (tensor space/functor) representations of the braid group derived from a meld of the (non-functor) Burau representation and the (functor) Deguchi {\em et al}-Kauffman--Saleur-Rittenberg representations here called Burau-Rittenberg representations. In its most supersymmetric case somewhat mystical cancellations of anomalies occur so that the Burau-Rittenberg representation extends to a loop Burau-Rittenberg representation. And this factors through $LH_n$. Let $SP_n$ denote the corresponding quotient algebra, $k$ the ground ring, and $t \in k$ the loop-Hecke parameter. We prove the following: 1) $LH_n$ is finite dimensional over a field. 2) The natural inclusion $LB_n \rightarrow LB_{n+1}$ passes to an inclusion $SP_n \rightarrow SP_{n+1}$. 3) Over $k=\mathbb{C}$, $SP_n / rad $ is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal's triangle. 4) We determine the other fundamental invariants of $SP_n$ representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A. 5) The structure of $SP_n $ is independent of the parameter $t$, except for $t= 1$. \item For $t^2 \neq 1$ then $LH_n \cong SP_n$ at least up to rank$n=7$ (for $t=-1$ they are not isomorphic for $n>2$; for $t=1$ they are not isomorphic for $n>1$). Finally we discuss a number of other intriguing points arising from this construction in topology, representation theory and combinatorics.
Each pointed topological space has an associated $π$-module, obtained from action of its first homotopy group on its second homotopy group. For the $3$-ball with a trivial link with $n$-components removed from its interior, its $π$-module $\mathcal{M}_n$ is of free type. In this paper we give an injection of the (extended) loop braid group into the group of automorphisms of $\mathcal{M}_n$. We give a topological interpretation of this injection, showing that it is both an extension of Artin's representation for braid groups and of Dahm's homomorphism for (extended) loop braid groups.
In this paper we compute a presentation for the group of ring motions of the split union of a Hopf link with Euclidean components and a Euclidean circle. A key part of this work is the study of a short exact sequence of groups of ring motions of general ring links in $\mathbb{R}^3$. This sequence allowed us to build the main result from the previously known case of the ring group with one component, which a particular case of the ring groups studied by Brendle and Hatcher. This work is a first step towards the computation of a presentation for groups of motions of H-trivial links with an arbitrary number of components.
In classical knot theory, Markov's theorem gives a way of describing all braids with isotopic closures as links in $\mathbb{R}^3$. We present a version of Markov's theorem for extended loop braids with closure in $B^3 \times S^1$, as a first step towards a Markov's theorem for extended loop braids and ribbon torus-links in $\mathbb{R}^4$.
Extended welded links are a generalization of Fenn, Rimányi, and Rourke's welded links. Their braided counterpart are extended welded braids, which are closely related to ribbon braids and loop braids. In this paper we prove versions of Alexander and Markov's theorems for extended welded braids and links, following Kamada's approach to the case of welded objects.
In this paper we introduce distinct approaches to loop braid groups, a generalisation of braid groups, and unify all the definitions that have appeared so far in literature, with a complete proof of the equivalence of these definitions. These groups have in fact been an object of interest in different domains of mathematics and mathematical physics, and have been called, in addition to loop braid groups, with several names such as of motion groups, groups of permutation-conjugacy automorphisms, braid-permutation groups, welded braid groups and untwisted ring groups. In parallel to this, we introduce an extension of these groups that appears to be a more natural generalisation of braid groups from the topological point of view. Throughout the text we motivate the interest in studying loop braid groups and give references to some of their applications.
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 particular their linearity.
Ribbon tangles are proper embeddings of tori and cylinders in the $4$-ball~$B^4$, "bounding" $3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathsf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathsf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau-Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathsf{C}ob_G$ over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones, and prove that the invariant $\mathsf{A}$ commutes with the compositions in this algebra. On the other hand, ribbon tangles admit diagrammatic representations, throught welded diagrams. We give a simple combinatorial description of $\mathsf{A}$ and of the algebra $\mathsf{C}ob_G$, and observe that our construction is a topological incarnation of the Alexander invariant of Archibald. When restricted to diagrams without virtual crossings, $\mathsf{A}$ provides a purely local description of the usual Alexander poynomial of links, and extends the construction by Bigelow, Cattabriga and the second author.