Unbraided wiring diagrams for Stein fillings of lens spaces

TL;DR

This paper provides an algorithm to draw unbraided wiring diagrams for Stein fillings of lens spaces, linking Lefschetz fibrations, symplectic disk arrangements, and cyclic quotient singularities, with applications to classifying Stein fillings.

Contribution

It introduces a new algorithm for unbraided wiring diagrams that represent Stein fillings, connecting Lefschetz fibrations, symplectic geometry, and singularity theory.

Findings

Wiring diagrams can be extended to symplectic disk arrangements in ^2.

The arrangement relates to the cyclic quotient singularity's plane curve germ.

A bijection between Stein fillings and Milnor fibers is explicitly constructed.

Abstract

In a previous work, we constructed a planar Lefschetz fibration on each Stein filling of any lens space equipped with its canonical contact structure. Here we describe an algorithm to draw an unbraided wiring diagram whose associated planar Lefschetz fibration obtained by the method of Plamenevskaya and Starkston, where the lens space with its canonical contact structure is viewed as the contact link of a cyclic quotient singularity, is equivalent to the Lefschetz fibration we constructed on each Stein filling of the lens space at hand. Coupled with the work of Plamenevskaya and Starkston, we obtain the following result as a corollary: The wiring diagram we describe can be extended to an arrangement of symplectic graphical disks in $\mathbb{C}^2$ with marked points, including all the intersection points of these disks, so that by removing the proper transforms of these disks from the blowup of $\mathbb{C}^2$ along those marked points one recovers the Stein filling along with the Lefschetz fibration. Moreover, the arrangement is related to the decorated plane curve germ representing the cyclic quotient singularity by a smooth graphical homotopy. As another application, we set up an explicit bijection between the Stein fillings of any lens space with its canonical contact structure, and the Milnor fibers of the corresponding cyclic quotient singularity, which was first obtained by N\'{e}methi and Popescu-Pampu, using different methods.