@article{e0f1b090989a4aa7968d897c8d390835,

title = "The Two-Eyes Lemma: A Linking Problem for Table-Top Necklaces",

abstract = "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.",

keywords = "Horoball, Hyperbolic, Packing, Spheres",

author = "David Gabai and Robert Meyerhoff and Andrew Yarmola",

note = "Funding Information: The first author was partially supported by National Science Foundation Grants DMS-1006553, DMS-1607374, and DMS-2003892. The second author was partially supported by National Science Foundation Grant DMS-1308642. The third author was partially supported as a Princeton VSRC with DMS-1006553. We thank the referee for a careful reading and suggested improvements, especially for the simplification of the argument in Step 4 of Lemma 2 , which originally relied on arguments from hyperbolic geometry. Funding Information: The first author was partially supported by National Science Foundation Grants DMS-1006553, DMS-1607374, and DMS-2003892. The second author was partially supported by National Science Foundation Grant DMS-1308642. The third author was partially supported as a Princeton VSRC with DMS-1006553. We thank the referee for a careful reading and suggested improvements, especially for the simplification of the argument in Step 4 of Lemma , which originally relied on arguments from hyperbolic geometry. Funding Information: The first author was partially supported by National Science Foundation Grants DMS-1006553, DMS-1607374, and DMS-2003892. The second author was partially supported by National Science Foundation Grant DMS-1308642. The third author was partially supported as a Princeton VSRC with DMS-1006553. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer Japan KK, part of Springer Nature.",

year = "2022",

month = apr,

doi = "10.1007/s00373-021-02439-x",

language = "English (US)",

volume = "38",

journal = "Graphs and Combinatorics",

issn = "0911-0119",

publisher = "Springer Japan",

number = "2",

}