Closures of T-homogeneous braids are real algebraic

Bode, Benjamin

doi:10.2140/agt.2025.25.1075

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Closures of T-homogeneous braids are real algebraic

Benjamin Bode

Algebraic & Geometric Topology 25 (2025) 1075–1115

DOI: 10.2140/agt.2025.25.1075

Abstract

A link in $S^{3}$ is called real algebraic if it is the link of an isolated singularity of a polynomial map from $ℝ^{4}$ to $ℝ^{2}$ . 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 both 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.

Keywords

real algebraic link, isolated singularity, semiholomorphic function, P-fibered braid, homogeneous braid, Rampichini diagram

Mathematical Subject Classification

Primary: 57K10

Secondary: 14P25, 32S55

References

Publication

Received: 4 April 2023

Revised: 4 November 2023

Accepted: 22 January 2024

Published: 16 May 2025

Authors

Benjamin Bode
Instituto de Ciencias Matemáticas CSIC Madrid Spain	Departamento de Matemática Aplicada a la Ingeniería Industrial Escuela Técnica Superior de Ingeniería y Diseño Industrial Universidad Politécnica de Madrid Madrid Spain
https://sites.google.com/view/benjaminbode

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Volume 25, issue 2 (2025)