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Band diagrams of
immersed surfaces in $4$-manifolds
Mark Hughes, Seungwon Kim and Maggie Miller |
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Algebraic & Geometric Topology 25 (2025) 1731–1791
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Abstract |
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We study immersed surfaces in smooth 4-manifolds via singular banded unlink diagrams. Such a diagram consists of a singular link with bands inside a Kirby diagram of the ambient 4-manifold, representing a level set of the surface with respect to an associated Morse function. We show that every self-transverse immersed surface in a smooth, orientable, closed 4-manifold can be represented by a singular banded unlink diagram, and that such representations are uniquely determined by the ambient isotopy or equivalence class of the surface up to a set of singular band moves which we define explicitly. By introducing additional finger, Whitney and cusp diagrammatic moves, we can use these singular band moves to describe homotopies or regular homotopies as well. Using these techniques, we introduce bridge trisections of immersed surfaces in arbitrary trisected 4-manifolds and prove that such bridge trisections exist and are unique up to simple perturbation moves. We additionally give some examples of how singular banded unlink diagrams may be used to perform computations or produce explicit homotopies of surfaces. |
Keywords
$4$-manifold, surface, band, diagram, knot, trisection
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Mathematical Subject Classification
Primary: 57K45
Secondary: 57K40
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References |
Publication
Received: 29 April 2023
Revised: 27 November 2023
Accepted: 12 May 2024
Published: 20 June 2025
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Authors |
| Mark Hughes | |
| Department of Mathematics Brigham Young University Provo, UT United States |
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| Seungwon Kim | |
| Sungkyunkwan University Suwon South Korea |
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| Maggie Miller | |
| Department of Mathematics The University of Texas at Austin Austin, TX United States |
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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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