Generalized Chain Surgeries and Applications

TL;DR

This paper explores Stein handlebody diagrams of Milnor fibers for Brieskorn singularities and investigates a symplectic operation involving exchanging Stein fillings, interpreted as a symplectic sum and monodromy substitution.

Contribution

It provides explicit Stein handlebody diagrams for Milnor fibers and introduces a novel symplectic operation based on exchanging Stein fillings, with dual interpretations.

Findings

Explicit Stein handlebody diagrams for Milnor fibers

A new symplectic operation via exchanging Stein fillings

Dual interpretations as symplectic sum and monodromy substitution

Abstract

We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this operation, one as a symplectic sum and the other as a monodromy substitution in a Lefschetz fibration.