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

TL;DR

This paper extends the concept of standard position for surfaces from classical alternating links in the 3-sphere to a broader class called weakly generalized alternating links, with applications in understanding their structure.

Contribution

It generalizes the standard position technique to weakly generalized alternating links, including those on higher genus surfaces and in various 3-manifolds.

Findings

All such links are prime.

Essential Conway spheres interact with the diagram as in classical case.

Standard position applies to a broader class of links.

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 non-split 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.