Real algebraic links in $S^3$ 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 $n-1$ distinct non-zero points in the complex plane. This results in an action of the braid group $\mathbb{B}_n$ on the set of $n$-adic integers $\mathbb{Z}_n$ for all natural numbers $n\geq 2$. 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 $B$ an infinite sequence of braids, whose braid types are invariants of $B$. We present computations for the cases of $n=2$ and $n=3$ and use these to show that an infinite family of braids close to real algebraic links, i.e., links of isolated singularities of real polynomials $\mathbb{R}^4\to\mathbb{R}^2$.