Combinatorial extension of a simple construction of Lefschetz fibrations

TL;DR

This paper extends a method for constructing Lefschetz fibrations by allowing variations in the fiber, enabling the creation of fibrations with different fibers but the same total space, including applications to knot traces.

Contribution

It introduces a combinatorial extension that varies the fiber in Lefschetz fibrations, providing new constructions for Stein surfaces and explicit comparisons with existing open books.

Findings

Constructed PALFs with genus 1 fibers for knot traces of Legendrian positive twist and torus knots.

Proved the existence of multiple PALFs with the same total space but different fibers.

Demonstrated the equivalence of their PALF with known open book decompositions for torus knots.

Abstract

In a previous work, we introduced a simple and systematic method for constructing a positive allowable Lefschetz fibration (PALF) from a 2-handlebody decomposition of a given Stein surface. In this paper, we present a combinatorial extension of this construction, focusing on the flexibility of the regular fiber. By introducing variations in the isotopy of the 0-handle during the construction process, we obtain PALFs whose total spaces are diffeomorphic to the original Stein surface but which possess different regular fibers. As a primary application, we prove the existence of PALFs with genus $1$ regular fibers whose total spaces are diffeomorphic to the knot traces of Legendrian positive twist knots and positive torus knots $T_{2, 2n+1}$. Furthermore, we explicitly compare our PALF associated with the positive torus knot $T_{2, 2n+1}$ to the specific open book decomposition generated by Avdek's Algorithm 2, demonstrating that the regular fiber and monodromy of our construction coincide with the page and monodromy of the corresponding open book.