Lefschetz fibrations on knot surgery $4$-manifolds via Stallings twist

TL;DR

This paper constructs families of knot surgery 4-manifolds with multiple nonisomorphic Lefschetz fibration structures sharing the same fiber genus, using Stallings twists on fibered knots to generate diverse monodromy groups.

Contribution

It introduces a method to produce 4-manifolds with multiple distinct Lefschetz fibrations via knot surgery and Stallings twists, expanding understanding of their monodromy structures.

Findings

Existence of arbitrarily many nonisomorphic Lefschetz fibrations on the same 4-manifold

Construction of fibered knots via Stallings twist from slice knots

Distinct monodromy groups for different Lefschetz fibrations

Abstract

In this article we construct a family of knot surgery $4$-manifolds admitting arbitrarily many nonisomorphic Lefschetz fibration structures with the same genus fiber. We obtain such families by performing knot surgery on an elliptic surface $E(2)$ using connected sums of fibered knots obtained by Stallings twist from a slice knot $3_1 \sharp 3^*_1$. By comparing their monodromy groups induced from the corresponding monodromy factorizations, we show that they admit mutually nonisomorphic Lefschetz fibration structures.