Isotopies of surfaces in 4-manifolds via banded unlink diagrams

Abstract

In this paper, 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 $\mathbb{C}{P}^2$ (i.e. spheres in the generating homology class), proving that many explicit unit surfaces are isotopic to the standard $\mathbb{C}{P}^1$. This strengthens some previously known results about the Gluck twist in $S^4$, related to Kirby problem 4.23.

arXiv comment39 pages, 21 figures. Corrected description of horizontal isotopy to include intersection with ascending manifolds of index-1 critical points, which induces slides over dotted circles. Also corrected various small typos and made other expositional edits

References in corpus (6)

• Trisecting 4-manifolds
• Bridge trisections of knotted surfaces in $S^4$
• Classification of trisections and the Generalized Property R Conjecture
• Bridge trisections of knotted surfaces in 4--manifolds
• All 2-dimensional links in 4-space live inside a universal 3-dimensional polyhedron
• Bridge and pants complexities of knots

Cited by in corpus (8)

• Gluck twisting roll spun knots
• Filling braided links with trisected surfaces
• Concordance of Surfaces and the Freedman-Quinn Invariant
• Unchaining surgery, branched covers, and pencils on elliptic surfaces
• Flattening knotted surfaces
• Branched covers and pencils on hyperelliptic Lefschetz fibrations
• Band diagrams of immersed surfaces in 4-manifolds
• Shadows of 2-knots and complexity