A Simple Construction of Lefschetz Fibrations on Compact Stein Surfaces

TL;DR

This paper introduces a straightforward method to construct positive allowable Lefschetz fibrations on compact Stein surfaces, providing smaller genus fibers and an alternative proof of existing results.

Contribution

It offers a simple construction technique for PALFs on Stein surfaces from 2-handlebody decompositions, improving fiber genus bounds and defining a new knot invariant.

Findings

Constructs PALFs with small genus fibers.

Provides an alternative proof for existing Stein surface results.

Defines a new knot invariant related to fiber genus.

Abstract

Loi-Piergallini, Akbulut-Ozbagci, and Akbulut-Arikan showed that every compact Stein surface admits a positive allowable Lefschetz fibration over the disk $D^2$ with bounded fibers (PALF in short), and they provided constructions of PALF's corresponding to compact Stein surfaces. In this paper, we present a simple method for constructing a PALF from a 2-handlebody decomposition of any given compact Stein surface. Our method yields PALF's whose regular fibers have small genus, and it provides an alternative constructive proof of the above result. We also define the minimal genus of a regular fiber of a PALF on the knot trace of a knot $K$ with framing one less than its maximal Thurston-Bennequin number as an invariant of $K$. When the grid number of $K$ is $N$, our construction produces a PALF whose regular fiber has genus at most $(N - 1)/2$, showing that the defined invariant is bounded above by $(N - 1)/2$.