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

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 fo...

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Bibliographic Details
Published in:arXiv.org 2023-02
Main Authors: Purcell, Jessica S, Tsvietkova, Anastasiia
Format: Article
Language:English
Subjects:
Combinatorial analysis
Links
Manifolds
Online Access:Get full text
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Summary: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 of 3-manifolds, namely weakly generalized alternating link complements. Such link complements include all classical prime non-split alternating links in 3-sphere, but also many complements of 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 in such links interact with the diagram exactly as in classical alternating links.
ISSN:2331-8422