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Closures of T-homogeneous braids are real algebraic

Benjamin Bode

Algebraic & Geometric Topology 25 (2025) 1075–1115
Abstract

A link in S3 is called real algebraic if it is the link of an isolated singularity of a polynomial map from 4 to 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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