The reduced Burau representation is a natural action of the braid group Bn on the first homology group H1(D̃n;Z) of a suitable infinite cyclic covering space D̃n of the n-punctured disc Dn. It is ...known that the Burau representation is faithful for n≤3 and that it is not faithful for n≥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++ program.
The High-Luminosity phase of the Large Hadron Collider at CERN (HL-LHC) poses stringent requirements on calorimeter performance in terms of resolution, pileup resilience and radiation hardness. A ...tungsten-CeF 3 sampling calorimeter is a possible option for the upgrade of current detectors. A prototype, read out with different types of wavelength-shifting fibers, has been built and exposed to high energy electrons, representative for the particle energy spectrum at HL-LHC, at the CERN SPS H4 beam line. This paper shows the performance of the prototype, mainly focussing on energy resolution and uniformity. A detailed simulation has been also developed in order to compare with data and to extrapolate to different configurations to be tested in future beam tests. Additional studies on the calorimeter and the R&D projects ongoing on the various components of the experimental setup will be also discussed.
Trans. A. Razmadze Math. Inst. 172 (2018) \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.
We introduce new operations reducing the number of Seifert circles in link diagrams of a special type. The operations are similar to one described in Mem. Amer. Math. Soc. 508 (1993) and Math. Proc. ...Cambridge Philos. Soc. 111 (2) (1992) 273. We discuss a conjecture about the number of Seifert circles that can be canceled by applying the operation repeatedly. We translate the problem into one belonging to graph theory.
TMJ, Special issue, 2021, 57-62 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
$\mathbb{Z}_pt,t^{-1}$ for any integer $p > 1$.
\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.
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 \(\mathbb{Z}_pt,t^{-1}\) for any integer \(p > 1\).
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