Standard position for surfaces in link complements in arbitrary 3-manifolds

Purcell, Jessica S.; Tsvietkova, Anastasiia

doi:10.2140/agt.2026.26.825

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Standard position for surfaces in link complements in arbitrary 3-manifolds

Jessica S. Purcell and Anastasiia Tsvietkova

Algebraic & Geometric Topology 26 (2026) 825–862

DOI: 10.2140/agt.2026.26.825

Abstract

Since the 1980s, it has been known that essential surfaces in alternating link complements can be isotoped to be transverse to the link diagram almost everywhere, with the exception of some well-understood intersections, and described combinatorially as a result. This was called standard position for surfaces and has had numerous applications. However, the original techniques only apply to classical alternating links projected onto the 2-sphere inside the 3-sphere. In this paper, we prove that standard position for surfaces can be extended to a broader class, namely weakly generalized alternating links. Such links include all classical prime nonsplit alternating links in the 3-sphere, and also many links that are alternating on higher-genus surfaces, or lie in manifolds besides the 3-sphere. As an application, we show that all such links are prime, and that under mild restrictions, essential Conway spheres for such links interact with the diagram exactly as in the classical alternating setting.

Keywords

knots, surfaces, 3-manifolds

Mathematical Subject Classification

Primary: 57K10, 57K12, 57K30

References

Publication

Received: 26 May 2022

Revised: 8 November 2024

Accepted: 27 February 2025

Published: 1 April 2026

Authors

Jessica S. Purcell
School of Mathematics Monash University Clayton, VIC Australia

Anastasiia Tsvietkova
Department of Mathematics and Computer Science Rutgers University, Newark Newark, NJ United States

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Volume 26, issue 3 (2026)