Closures of T-homogeneous braids are real algebraic

TL;DR

This paper proves that closures of T-homogeneous braids are real algebraic links, providing explicit polynomial constructions and advancing understanding of the relationship between algebraic and fibered links.

Contribution

It establishes that closures of T-homogeneous braids are real algebraic links and constructs explicit semiholomorphic polynomial maps for them.

Findings

Closures of T-homogeneous braids are real algebraic links.

Constructs explicit polynomial maps as semiholomorphic functions.

Provides bounds on polynomial degrees for these maps.

Abstract

A link in $S^3$ is called real algebraic if it is the link of an isolated singularity of a polynomial map from $\mathbb{R}^4$ to $\mathbb{R}^2$. It is known that every real algebraic link is fibered and it is conjectured that the converse is also true. We prove this conjecture for a large family of fibered links, which includes closures of T-homogeneous (and therefore also homogeneous) braids and braids that can be written as a product of the dual Garside element and a positive word in the Birman-Ko-Lee presentation. The proof offers a construction of the corresponding real polynomial maps, which can be written as semiholomorphic functions. We obtain information about their polynomial degrees.