Unbraided wiring diagrams for Stein fillings of lens spaces

TL;DR

The paper provides an explicit, combinatorial bridge between Stein fillings of lens spaces and Milnor fibres of associated cyclic quotient singularities. By developing an algorithm to construct unbraided wiring diagrams that encode planar Lefschetz fibrations, it shows how the vanishing cycles of these fibrations correspond to a wiring-diagram framework derived from blowups, twists, and decorations of plane curve germs. This yields a direct bijection between Lisca’s Stein fillings and Milnor fibres, clarifies the relation to decorated curve germs and Scott deformations, and furnishes practical tools (incidence matrices and graphical disk arrangements) for distinguishing fillings and understanding their Lefschetz fibrations. The results unify symplectic, contact, and singularity-theoretic perspectives, with explicit constructions that extend to graphical disk arrangements and Milnor fibre models. The work advances explicit classification and constructive understanding of the interplay between contact lens spaces, planar Lefschetz fibrations, and singularity theory.

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 Némethi and Popescu-Pampu, using different methods.

Figures (32)

• Figure 1: From Lisca to de Jong and van Straten
• Figure 2: An example of an unbraided wiring diagram without any free points.
• Figure 3: Convex curves in $D_5$ assigned to the marked points in Figure \ref{['fig: wire']}.
• Figure 4: Starting from $\delta(x_4)$, we apply a counterclockwise half-twist on the subdisk enclosed by the dotted curve, at each step, going from left to right.
• Figure 5: Starting from $\delta(x_3)$, we apply a counterclockwise half-twist on the subdisk enclosed by the dotted curve, at each step, going from left to right.

Theorems & Definitions (36)

• Theorem 1
• Corollary 2
• Corollary 3
• Corollary 4
• Corollary 5
• Definition 6
• Definition 7
• Definition 8
• Definition 9
• Definition 10