Positive Dehn Twist Expressions for some New Involutions in Mapping Class Group

TL;DR

This paper introduces new positive Dehn twist expressions for certain involutions in the mapping class group, leading to new symplectic Lefschetz fibrations and their signature computations.

Contribution

It provides novel words in the mapping class group representing involutions, expanding the class of known symplectic Lefschetz fibrations.

Findings

New positive Dehn twist expressions for involutions

Construction of new symplectic Lefschetz fibrations

Computed signatures of the constructed fibrations

Abstract

The well-known fact that any genus $g$ symplectic Lefschetz fibration $ X^{4}\to S^{2}$ is given by a word that is equal to the identity element in the mapping class group and each of whose elements is given by a positive Dehn twist, provides an intimate relationship between words in the mapping class group and 4-manifolds that are realized as symplectic Lefschetz fibrations. In this article we provide new words in the mapping class group, hence new symplectic Lefschetz fibrations. We also compute the signatures of those symplectic Lefschetz fibrations.