On Positive Braids, Monodromy Groups and Framings

TL;DR

This paper introduces a new braid monodromy group for positive braids, linking it to framed mapping class groups and providing a knot invariant, with applications to singularity theory.

Contribution

It generalizes the geometric monodromy group to positive braids and establishes a connection with framed mapping class groups, leading to a new knot invariant.

Findings

The braid monodromy group is well-defined for positive braids.

For knot closures, the group matches a framed mapping class group.

The geometric monodromy of an irreducible singularity is determined by genus and Arf invariant.

Abstract

We associate to every positive braid a braid monodromy group, generalizing the geometric monodromy group of an isolated plane curve singularity. If the closure of the braid is a knot, we identify the corresponding group with a framed mapping class group. In particular, this gives a well defined knot invariant. As an application, we obtain that the geometric monodromy group of an irreducible singularity is determined by the genus and the Arf invariant of the associated knot.