|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
Bridge trisections and
Seifert solids
Jason Joseph, Jeffrey Meier, Maggie Miller and Alexander Zupan |
|
Algebraic & Geometric Topology 25 (2025) 1501–1522
|
 |
Abstract |
|
We adapt Seifert’s algorithm for classical knots and links to the setting of triplane diagrams for bridge trisected surfaces in the -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 -handles; in contrast, we give examples of Seifert solids (not coming from our construction) whose exteriors require arbitrarily many -handles. We conclude with two classification results. The first shows that surfaces admitting doubly standard shadow diagrams are unknotted. The second says that a -bridge trisection in which some sector contains at least patches is completely decomposable, thus the corresponding surface is unknotted. This settles affirmatively a conjecture of the second and fourth authors. |
Keywords
knot theory, surface, trisection, bridge trisection,
2-knot, Seifert surface, Seifert solid
|
Mathematical Subject Classification
Primary: 57K45
Secondary: 57K10
|
References |
Publication
Received: 4 November 2022
Revised: 28 August 2023
Accepted: 23 October 2023
Published: 20 June 2025
|
Authors |
| Jason Joseph | |
| Department of Mathematics North Carolina School of Science and Mathematics Morganton, NC United States |
|
| Jeffrey Meier | |
| Department of Mathematics Western Washington University Bellingham, WA United States |
|
| Maggie Miller | |
| Department of Mathematics University of Texas at Austin Austin, TX United States |
|
| Alexander Zupan | |
| Department of Mathematics University of Nebraska–Lincoln Lincoln, NE United States |
|
| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
Open Access made possible by participating institutions via Subscribe to Open.
