Stable soap bubble clusters with multiple torus bubbles

Abstract: In the last two centuries and more particularly in the last decades, the geometry of foams has become an important research domain, in mathematics, physics, material sciences and biology. Most of the simplest geometrical observations of bubble clusters have long resisted rigorous mathematical proofs. Geometries can even get more complicated if immiscible fluids are considered. Although they have to fulfill Plateau's laws like soap bubble clusters if the surface tensions are close to unity, this is not the case in general. In 1996, Frederick J. Almgren asked whether there is "any stable cluster of bubbles in $\mathbb{R}<sup>3$ with some bubble being topologically a torus". We propose to answer the latter numerically with simple numerical examples. We build stable soap bubble clusters with a triple torus bubble, a fivefold torus bubble or an elevenfold torus bubble. The construction uses the geometry of a simple immiscible fluids cluster with a torus bubble.