#### Volume 24, issue 3 (2020)

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Isotopies of surfaces in $4$–manifolds via banded unlink diagrams

### Mark C Hughes, Seungwon Kim and Maggie Miller

Geometry & Topology 24 (2020) 1519–1569
##### Abstract

We study surfaces embedded in $4$–manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary $4$–manifold. This extends work of Swenton and Kearton–Kurlin in ${S}^{4}$. As an application, we show that bridge trisections of isotopic surfaces in a trisected $4$–manifold are related by a sequence of perturbations and deperturbations, affirmatively proving a conjecture of Meier and Zupan. We also exhibit several isotopies of unit surfaces in $ℂ\phantom{\rule{-0.17em}{0ex}}{P}^{2}$ (ie spheres in the generating homology class), proving that many explicit unit surfaces are isotopic to the standard $ℂ\phantom{\rule{-0.17em}{0ex}}{P}^{1}$. This strengthens some previously known results about the Gluck twist in ${S}^{4}$, related to Kirby problem 4.23.

##### Keywords
$4$–manifold, knot, surface, diagram
Primary: 57K45
Secondary: 57K40
##### Publication
Received: 20 February 2019
Revised: 29 March 2020
Accepted: 23 May 2020
Published: 30 September 2020
Proposed: András I Stipsicz
Seconded: Mladen Bestvina, Ciprian Manolescu
##### Authors
 Mark C Hughes Department of Mathematics Brigham Young University Provo, UT United States https://math.byu.edu/~hughes/ Seungwon Kim Center for Geometry and Physics Institute for Basic Science (IBS) Pohang South Korea https://sites.google.com/view/seungwonkim Maggie Miller Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States https://web.math.princeton.edu/~maggiem