Bridge trisections and Seifert solids

TL;DR

This paper extends Seifert's algorithm to bridge trisections of surfaces in the 4-sphere, constructing Seifert solids via Heegaard diagrams and providing classification results for unknotted surfaces.

Contribution

It introduces a novel adaptation of Seifert's algorithm for bridge trisections and classifies unknotted surfaces under certain conditions.

Findings

Seifert solids can be constructed with exteriors without 3-handles.

Surfaces with doubly-standard shadow diagrams are unknotted.

Certain b-bridge trisections are completely decomposable, implying unknottedness.

Abstract

We adapt Seifert's algorithm for classical knots and links to the setting of tri-plane diagrams for bridge trisected surfaces in the 4-sphere. Our approach allows for the construction of a Seifert solid that is described by a Heegaard diagram. The Seifert solids produced can be assumed to have exteriors that can be built without 3-handles; in contrast, we give examples of Seifert solids (not coming from our construction) whose exteriors require arbitrarily many 3-handles. We conclude with two classification results. The first shows that surfaces admitting doubly-standard shadow diagrams are unknotted. The second says that a $b$-bridge trisection in which some sector contains at least $b-1$ patches is completely decomposable, thus the corresponding surface is unknotted. This settles affirmatively a conjecture of the second and fourth authors.