How do curved spheres intersect in 3-space?

TL;DR

This paper investigates whether two unions of disjoint circles on spheres can be realized as intersections of two 3D spheres with specific arrangements, providing a counterexample and a condition for such configurations.

Contribution

It proves that not all circle unions can be realized as intersections of spheres and establishes a necessary and sufficient condition for realizability.

Findings

Counterexample showing some unions cannot be realized as sphere intersections

A necessary and sufficient condition for realizability

A graph-based restatement enabling brute-force checking

Abstract

The following problem was proposed in 2010 by S. Lando. Let $M$ and $N$ be two unions of the same number of disjoint circles in a sphere. Do there always exist two spheres in 3-space such that their intersection is transversal and is a union of disjoint circles that is situated as $M$ in one sphere and as $N$ in the other? Union $M'$ of disjoint circles is {\it situated} in one sphere as union $M$ of disjoint circles in the other sphere if there is a homeomorphism between these two spheres which maps $M'$ to $M$. We prove (by giving an explicit example) that the answer to this problem is "no". We also prove a necessary and sufficient condition on $M$ and $N$ for existing of such intersecting spheres. This result can be restated in terms of graphs. Such restatement allows for a trivial brute-force algorithm checking the condition for any given $M$ and $N$. It is an open question if a faster algorithm exist.