The Two-Eyes Lemma: A Linking Problem for Table-Top Necklaces

In this note, we answer a combinatorial question that is inspired by cusp geometry of hyperbolic 3-manifolds. A table-top necklace is a collection of sequentially tangent beads (i.e. spheres) with disjoint interiors lying on a flat table (i.e. a plane) such that each bead is of diameter at most one and is tangent to the table. We analyze the possible configurations of a necklace with at most 8 beads linking around two other spheres whose diameter is exactly 1. We show that all the beads are forced to have diameter one, the two linked spheres are tangent, and that each bead must be tangent to at least one of the two linked spheres. In fact, there is a 1-parameter family of distinct configurations.