Real algebraic links in S3 and braid group actions on the set of n-adic integers

Abstract

We construct an infinite tower of covering spaces over the configuration space of [Formula: see text] distinct nonzero points in the complex plane. This results in an action of the braid group [Formula: see text] on the set of [Formula: see text]-adic integers [Formula: see text] for all natural numbers [Formula: see text]. We study some of the properties of these actions such as continuity and transitivity. The construction of the actions involves a new way of associating to any braid [Formula: see text] an infinite sequence of braids, whose braid types are invariants of [Formula: see text]. We present computations for the cases of [Formula: see text] and [Formula: see text] and use these to show an infinite family of braids close to real algebraic links, i.e. links of isolated singularities of real polynomials [Formula: see text].