# Constructing links of isolated singularities of polynomials $\mathbb{R}^4\to\mathbb{R}^2$

## TL;DR

This paper extends previous work to construct real polynomial functions with isolated singularities whose links are closures of certain braids, demonstrating their real algebraic nature and explicit fibration properties.

## Contribution

It introduces a method to construct polynomials with isolated singularities linked to specific braid closures, including squares of homogeneous braids and lemniscate links, satisfying the strong Milnor condition.

## Key findings

- Closures of certain braids are shown to be real algebraic links.
- Constructed polynomials satisfy the strong Milnor condition.
- Explicit fibrations of the link complements are provided.

## Abstract

We show that if a braid $B$ can be parametrised in a certain way, then previous work can be extended to a construction of a polynomial $f:\mathbb{R}^4\to\mathbb{R}^2$ with the closure of $B$ as the link of an isolated singularity of $f$, showing that the closure of $B$ is real algebraic. In particular, we prove that closures of squares of strictly homogeneous braids and certain lemniscate links are real algebraic. We also show that the constructed polynomials satisfy the strong Milnor condition, providing an explicit fibration of the complement of the closure of $B$ over $S^1$.