Doubly pointed trisection diagrams and surgery on 2-knots

TL;DR

This paper introduces a method to represent 2-knots in 4-manifolds using doubly pointed trisection diagrams, proving their uniqueness up to stabilization and handleslides, and details how to perform classical surgeries directly from these diagrams.

Contribution

It develops a new diagrammatic approach for 2-knot surgeries in 4-manifolds, enabling direct visualization and manipulation of complex cut-and-paste operations.

Findings

Doubly pointed trisection diagrams are unique up to stabilization and handleslides.

Methods to derive diagrams for classical surgeries from doubly pointed diagrams.

Illustrations of surgeries like Gluck, blowdown, and rational blowdown with examples.

Abstract

We study embedded spheres in 4-manifolds (2-knots) via doubly pointed trisection diagrams, showing that such descriptions are unique up to stabilization and handleslides, and we describe how to obtain trisection diagrams for certain cut-and-paste operations along 2-knots directly from doubly pointed trisection diagrams. The operations described are classical surgery, Gluck surgery, blowdown, and $(\pm4)$-rational blowdown, and we illustrate our techniques and results with many examples.