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 (CCBY 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 braidtheoretic 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 3braids. The main part of the thesis is concerned with the socalled 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:  02 Aug 2023 11:16 
URI:  https://boristheses.unibe.ch/id/eprint/4461 
Actions (login required)
View Item 