Related papers: Hurwitz stabilisers of some short redundant Artin …

For a finite group $G$, the Hurwitz space $\mathcal{H}^{in}_{r,g}(G)$ is the space of genus $g$ covers of the Riemann sphere with $r$ branch points and the monodromy group $G$. In this paper, we give a complete list of primitive genus one…

Let n be greater than or equal to 3. We study the quotient group B\_n/[P n,P\_n] of the Artin braid group B\_n by the commutator subgroup of its pure Artin braid group P\_n. We show that B\_n/[P n,P\_n] is a crystallographic group, and in…

The reduced Burau representation $V_n$ of the braid group $B_n$ is obtained from the action of $B_n$ on the homology of an infinite cyclic cover of the $n$-punctured disc. In this note, we calculate $H_*(B_n;V_n)$ as a module over the…

In this paper we present recent results on the computation of skein modules of 3-manifolds using braids and appropriate knot algebras. Skein modules generalize knot polynomials in $S^3$ to knot polynomials in arbitrary 3-manifolds and they…

We establish a strong, geometric lower bound on the (sequential) topological complexity of the unordered configuration spaces of a general graph. As an application, we show that, for most graphs, the topological complexity eventually…

We prove that the moduli spaces A_3(D) of polarized abelian threefolds with polarizations of types D=(1,1,2), (1,2,2), (1,1,3) or (1,3,3) are unirational. The result is based on the study of families of simple coverings of elliptic curves…

In this paper we prove homological stability for certain subgroups of surface braid groups. Alternatively, this is equivalent to proving homological stability for configurations of subsets of exactly $\xi$ points in a surface as we increase…

For a tree $G$, we study the changing behaviors in the homology groups $H_i(B_nG)$ as $n$ varies, where $B_nG := \pi_1($UConf$_n(G))$. We prove that the ranks of these homologies can be described by a single polynomial for all $n$, and…

We solve the Hurwitz monodromy problem for degree-4 covers. That is, the Hurwitz space H_{4,g} of all simply branched covers of P^1 of degree 4 and genus g is an unramified cover of the space P_{2g+6} of (2g+6)-tuples of distinct points in…

This article gives certain fibre bundles associated to the braid groups which are obtained from a translation as well as conjugation on the complex plane. The local coefficient systems on the level of homology for these bundles are given in…

We define an action of Artin's braid group on a finite dimensional algebra.

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic from the braid group to the mapping class group. We prove here that this map is trivial in stable homology with…

Motivated by the problem of Hurwitz equivalence of $\Delta ^2$ factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, obtained by projecting the $\Delta ^2$ factorizations into $S_n$. We get…

To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artin's braid group B_n. Using Hansen's polynomial covering space theory, we give a new interpretation of…

We study the action of the group of polynomial automorphisms of C^n (n>2) which preserve the Markoff-Hurwitz polynomial H(x):= x_1^2 + x_2^2 + ... + x_n^2 - x_1 x_2 ... x_n. Our main results include the determination of the group, the…

Some curious structural similarities between a recent braid- and Hurwitz algebraic description of the unbroken internal symmetries for a single generations of Standard Model fermions were recently identified. The non-trivial braid groups…

A complete description of the Alexander modules of knotted $n$-manifolds in the sphere $S^{n+2}$, $n\geq 2$, and irreducible Hurwitz curves is given. This description is applied to investigate properties of the first homology groups of…

E. Kani has shown that the Hurwitz functor, which parametrizes the (normalized) genus 2 covers of degree 3 of an elliptic curve, is representable. In this paper the corresponding moduli scheme and the universal family are explicitly…

Some Kharitonov-like robust Hurwitz stability criteria are established for a class of complex polynomial families with nonlinearly correlated perturbations. These results are extended to the polynomial matrix case and non-interval…

We show that most of the genus-zero subgroups of the braid group $\mathbb{B}_3$ (which are roughly the braid monodromy groups of the trigonal curves on the Hirzebruch surfaces) are irrelevant as far as the Alexander invariant is concerned:…