Closures of T-homogeneous braids are real algebraic

Bode, Benjamin¹

¹Instituto 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

Algebraic and Geometric Topology, Tome 25 (2025) no. 2, pp. 1075-1115

Résumé

A link in S3 is called real algebraic if it is the link of an isolated singularity of a polynomial map from ℝ4 to ℝ2. 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.

Détail

DOI : 10.2140/agt.2025.25.1075

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

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