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

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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.

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ArticleOpen access14 September 2020

ArticleOpen access06 July 2021

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Acknowledgements

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 Lemma2, which originally relied on arguments from hyperbolic geometry.

Funding

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.

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    David Gabai & Andrew Yarmola

  2. Math Department, Maloney Hall, Fifth Floor, 140 Commonwealth Avenue, Chestnut Hill, MA, 02467, USA

    Robert Meyerhoff

Authors
  1. David Gabai

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  2. Robert Meyerhoff

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  3. Andrew Yarmola

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Correspondence toAndrew Yarmola.

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Gabai, D., Meyerhoff, R. & Yarmola, A. The Two-Eyes Lemma: A Linking Problem for Table-Top Necklaces.Graphs and Combinatorics38, 29 (2022). https://doi.org/10.1007/s00373-021-02439-x

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