Reducible braids and Garside theory

We show that reducible braids which are, in a Garside-theoretical sense, as simple as possible within their conjugacy class, are also as simple as possible in a geometric sense. More precisely, if a braid belongs to a certain subset of its conjugacy class which we call the stabilized set of sliding circuits, and if it is reducible, then its reducibility is geometrically obvious: it has a round or almost round reducing curve. Moreover, for any given braid, an element of its stabilized set of sliding circuits can be found using the well-known cyclic sliding operation. This leads to a polynomial time algorithm for deciding the Nielsen-Thurston type of any braid, modulo one well-known conjecture on the speed of convergence of the cyclic sliding operation.Comment: 28 pages, 4 figure

Twisted conjugacy in braid groups

In this note we solve the twisted conjugacy problem for braid groups, i.e. we propose an algorithm which, given two braids $u,v\in B_n$ and an automorphism $\phi \in Aut (B_n)$, decides whether $v=(\phi (x))^{-1}ux$ for some $x\in B_n$. As a corollary, we deduce that each group of the form $B_n \rtimes H$, a semidirect product of the braid group $B_n$ by a torsion-free hyperbolic group $H$, has solvable conjugacy problem

On the centralizer of generic braids

We study the centralizer of a braid from the point of view of Garside theory, showing that generically a minimal set of generators can be computed very efficiently, as the ultra summit set of a generic braid has a very particular structure. We present an algorithm to compute the centralizer of a braid whose generic-case complexity is quadratic on the length of the input, and which outputs a minimal set of generators in the generic case.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona

A geometric description of the extreme Khovanov cohomology

We prove that the hypothetical extreme Khovanov cohomology of a link is the cohomology of the independence simplicial complex of its Lando graph. We also provide a family of knots having as many non-trivial extreme Khovanov cohomology modules as desired, that is, examples of H-thick knots which are as far of being H-thin as desired.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona

TCAD Simulations and Characterization of High-Voltage Monolithic Active Pixel Sensors

High- Voltage Monolithic Active Pixel Sensors (HV-MAPS) have emerged as a promising technology for silicon tracking detectors in particle physics. HV-MAPS, selected as the foundational technology for the Mu3e Pixel Tracker and under investigation for potential implementation in future detector applications, presents unique design challenges due to its intricate structure and complex electric field distribution. This thesis presents the first comprehensive comparison of Technology Computer-Aided Design (TCAD) simulations and experimental measurements in HV-MAPS. The results show that the simulations correctly describe key experimental parameters like breakdown voltage and explain the loss of hit detection efficiency at the edges and corners of the pixels. The TCAD simulations provide insights into the behavior of the charge collection diode of MuPix8, ALTASPix, and MuPix10 prototypes, facilitating design optimizations. These studies primarily investigated the depletion zone, breakdown voltage and electric field distribution. Additionally, the characterization of MuPix10, using testbeam results, allows for the investigation of the efficiency and cluster size for different angles of incidence of the beam Furthermore, this research examines the impact of diffusion and drift on efficiency and cluster size for different voltage, resistivity, and thickness configurations. The findings of this investigation contribute to an enhanced understanding of HV-MAPS and their potential for developing more efficient and reliable silicon tracking detectors in particle physics experiments