Ferretti, Livio Clemente Emilio (2023). On positive braids and monodromy groups of plane curve singularities. (Thesis). Universität Bern, Bern
|
Text
23ferretti_l.pdf - Thesis Available under License Creative Commons: Attribution (CC-BY 4.0). Download (967kB) | Preview |
Abstract
This thesis is situated at the intersection of knot theory, singularity theory and surface topology. The first objects of consideration are secondary braid groups of positive braids, a conjectural braid-theoretic generalization of the local fundamental group of the discriminant complement of an isolated plane curve singularity. In particular, we prove that the secondary braid group is a link invariant for positive 3-braids. The main part of the thesis is concerned with the so-called monodromy group of a positive braid. First, we prove that such groups are a generalization of the geometric monodromy group of an isolated plane curve singularity. We then study those monodromy groups using the theory of framed mapping class groups. In particular, we prove that, for a prime positive braid whose closure is a knot of braid index at least 3, up to finitely many exceptions the monodromy group is a framed mapping class group. We then deduce that, under the same assumptions, the monodromy group is a knot invariant, determined by the genus and Arf invariant of the braid closure. Applied to singularity theory, this implies that the geometric monodromy group of an irreducible singularity not of type Aₙ is, up to finitely many exceptions, determined by the genus and Arf invariant of the associated knot. Finally, we briefly consider the extension of such techniques to more general fibred knots, obtained by plumbing Hopf bands. We obtain analogous results concerning monodromy groups of arborescent Hopf plumbings.
| Item Type: | Thesis |
|---|---|
| Dissertation Type: | Single |
| Date of Defense: | 29 June 2023 |
| Subjects: | 500 Science > 510 Mathematics |
| Institute / Center: | 08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics |
| Depositing User: | Sarah Stalder |
| Date Deposited: | 31 Jul 2023 14:10 |
| Last Modified: | 29 Jun 2024 22:25 |
| URI: | https://boristheses.unibe.ch/id/eprint/4461 |
Actions (login required)
 |
View Item |