Depicting a generalized shift move in crown diagrams

TL;DR

This paper introduces a diagrammatic method for generalized shift moves in crown diagrams of smooth 4-manifolds, simplifying equivalence proofs and aiding in the conversion of Lefschetz fibrations into crown diagrams.

Contribution

It provides a new diagrammatic approach to generalized shift moves, with applications to proving slide-equivalence and converting Lefschetz fibrations into crown diagrams.

Findings

Generalized shift moves preserve slide-equivalence of crown diagrams.

Method to convert genus g Lefschetz fibrations into crown diagrams.

Vanishing cycles are slide-equivalent to standard inclusions.

Abstract

This paper gives a diagrammatic way to perform a generalized shift move on a crown diagram of a smooth 4-manifold. Applications include a simplified proof that if two crown diagrams are related by a generalized shift move, then they are slide-equivalent; a method for converting a genus g > 1 Lefschetz fibration into a crown diagram; and the fact that the vanishing cycles of such a crown diagram are slide-equivalent to a standard inclusion of the Lefschetz vanishing cycles into a genus g + 1 surface.