arXiv subjects

A. Beridze

Publications and source records attributed to A. Beridze.

4 recordsLinked to original sources

arXiv2023

On the subgroup of $B_4$ that contains the kernel of Burau representation

It is known that there are braids $α$ and $β$ in the braid group $B_4$, such that the group $\langle α, β\rangle$ is a fee subgroup \cite{7}, which contains the kernel $K$ of the Burau map $ρ_4 : B_4 \to G L\left(3, \mathbb{Z}[t,t^{-1}]\right)$ \cite{6}, \cite{4}. In this paper we will prove that $K$ is subgroup of $G=\langle τ, Δ\rangle $, where $τ$ and $Δ$ are fourth and square roots of the generator $θ$ of the center $Z$ of the group $B_4$. Consequently, we will write elements of $K$ in terms of $τ^i,~~i=1,2,3$ and $Δ$. Moreover, we will show that the quotient group $G/Z$ is isomorphic to the free product $Z_4 *Z_2$.

math.GR↗

arXiv2021

Strong Shape Theory of Continuous Maps

The work is motivated by the papers [Ba1], [Ba2], [Ba7], [Ba11], [Be] and [Be-Tu]. In particular, the strong homology groups of continuous maps were defined and studied in [Be] and [Be-Tu]. To show that given groups are homology type functor, it was required to construct the corresponding shape category. In this paper, we study this very problem. In particular, using the methods developed in [Ba7], [Ba-Ts]. the strong shape theory of continuous maps of compact metric spaces, so-called strong fiber shape theory is constructed.

math.AT↗

arXiv2020

On the Burau Representation of $B_4$ modulo $p$

The problem of faithfulness of the (reduced) Burau representation for $n =4$ is known to be equivalent to the problem of whether certain two matrices $A$ and $B$ generate a free group of rank two. It is known that $A^3$ and $B^3$ generate a free group of rank two \cite{9}, \cite{10}, \cite{4}. We prove that they also generate a free group when considered as matrices over the $\mathbb{Z}_p[t,t^{-1}]$ for any integer $p > 1$.

math.GT↗

arXiv2018

Forks, Noodles and the Burau representation for $n=4$

\begin{abstract} The reduced Burau representation is a natural action of the braid group $B_n$ on the first homology group $H_1({\tilde{D}}_n;\mathbb{Z})$ of a suitable infinite cyclic covering space ${\tilde{D}}_n$ of the $n$--punctured disc $D_n$. It is known that the Burau representation is faithful for $n\le 3$ and that it is not faithful for $n\ge 5$. We use forks and noodles homological techniques and Bokut--Vesnin generators to analyze the problem for $n=4$. We present a Conjecture implying faithfulness and a Lemma explaining the implication. We give some arguments suggesting why we expect the Conjecture to be true. Also, we give some geometrically calculated examples and information about data gathered using a C\texttt{++} program.

math.GT↗