Constructing a broken Lefschetz fibration of S^4 with a spun or twist-spun torus knot fiber

TL;DR

This paper provides an explicit construction of broken Lefschetz fibrations on the 4-sphere with spun or twist-spun torus knot fibers, expanding the set of known examples in 4-manifold topology.

Contribution

It introduces a method to explicitly construct broken Lefschetz fibrations on S^4 with specific knot fibers, which previously lacked explicit examples.

Findings

Constructed fibrations have no cusps or Lefschetz singularities.

Provided explicit procedures for spun or twist-spun torus knot fibers.

Extended the class of known broken Lefschetz fibrations on S^4.

Abstract

Much work has been done on the existence and uniqueness of broken Lefschetz fibrations such as those by Auroux et al., Gay and Kirby, Lekili, Akbulut and Karakurt, Baykur, and Williams, but there has been a lack of explicit examples. A theorem of Gay and Kirby suggests the existence of a broken Lefschetz fibration of S^4 over S^2 with a 2-knot fiber. In the case of a spun or twist-spun torus knot, we present a procedure to construct such fibrations explicitly. The fibrations constructed have no cusps nor Lefschetz singularities.