Unbraided wiring diagrams for Stein fillings of lens spaces

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.