Principal analytic link theory in homology sphere links

TL;DR

This paper investigates the existence of knots in homology sphere links of surface singularities that correspond to zero sets of analytic germs, revealing a classification related to Brieskorn spheres.

Contribution

It characterizes when such knots exist in integral homology sphere links, specifically excluding certain Brieskorn spheres, linking knot theory with complex surface singularities.

Findings

Existence of knots linked to analytic germs depends on the topology of the homology sphere.

Such knots exist in all integral homology spheres except three specific Brieskorn spheres.

Provides a topological criterion for the analytic realization of knots in surface singularity links.

Abstract

For the link $M$ of a normal complex surface singularity $(X,0)$ we ask when a knot $K\subset M$ exists for which the answer to whether $K$ is the link of the zero set of some analytic germ $(X,0)\to (\mathbb C,0)$ affects the analytic structure on $(X,0)$. We show that if $M$ is an integral homology sphere then such a knot exists if and only if $M$ is not one of the Brieskorn homology spheres $M(2,3,5)$, $M(2,3,7)$, $M(2,3,11)$.