Milnor fibration, A'Campo's divide and Turaev's shadow

TL;DR

This paper introduces a method to construct shadowed polyhedra from divides, linking them to Lefschetz fibrations and fibered links with positive monodromy, advancing the understanding of 4-manifolds and their fibrations.

Contribution

It provides a novel construction technique connecting divides, shadowed polyhedra, and Lefschetz fibrations, demonstrating their equivalence and applications to fibered links.

Findings

Shadowed polyhedron from divides satisfies LF-property.

Lefschetz fibration of constructed polyhedron matches that of the divide.

Certain free divides produce fibered links with positive monodromy.

Abstract

We give a method for constructing a shadowed polyhedron from a divide. The 4-manifold reconstructed from a shadowed polyhedron admits the structure of a Lefschetz fibration if it satisfies a certain property, which we call the LF-property. We will show that the shadowed polyhedron constructed from a divide satisfies this property and the Lefschetz fibration of this polyhedron is isomorphic to the Lefschetz fibration of the divide. Furthermore, applying the same technique to certain free divides we will show that the links of those free divides are fibered with positive monodromy.