An Algorithm taking Kirby diagrams to Trisection diagrams

TL;DR

This paper introduces an algorithm that converts Kirby diagrams into trisection diagrams for closed oriented 4-manifolds, expanding the toolkit for visualizing and analyzing these manifolds, including non-orientable cases.

Contribution

The paper presents a novel algorithm for transforming Kirby diagrams into trisection diagrams, applicable to both orientable and non-orientable 4-manifolds.

Findings

Generates numerous examples of trisection diagrams from known Kirby diagrams.

Extends the algorithm to non-orientable 4-manifolds.

Provides a new method for visualizing 4-manifolds.

Abstract

We present an algorithm taking a Kirby diagram of a closed oriented $4$-manifold to a trisection diagram of the same manifold. This algorithm provides us with a large number of examples for trisection diagrams of closed oriented $4$-manifolds since many Kirby-diagrammatic descriptions of closed oriented $4$-manifolds are known. That being said, the algorithm does not necessarily provide particularly efficient trisection diagrams. We also extend this algorithm to work for the non-orientable case.