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FROM DOMINOES TO HEXAGONS

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📜 Abstract

There is a natural generalization of domino tilings to tilings of a polygon by hexagons, or, dually, configurations of oriented curves that meet in triples. We show exactly when two such tilings can be connected by a series of moves analogous to the domino flip move. The triple diagrams that result have connections to Legendrian knots, cluster algebras, and planar algebras.

✨ Summary

Main contribution

The paper introduces triple diagrams, a dual description of polygonal tilings by hexagons. These diagrams consist of oriented strands whose intersections involve three strands at a time, with orientations compatible with a checkerboard coloring of the complementary regions. Domino tilings arise as a special case after adding vertices to the long edges of dominoes.

The central combinatorial results are:

  • Every pairing of alternating incoming and outgoing boundary points can be realized by a triple diagram without closed strands.
  • Any two minimal triple diagrams with the same boundary matching are connected by a sequence of local 2 ↔ 2 moves, the analogue of the domino flip.
  • An arbitrary triple diagram can be reduced to a minimal one using 2 ↔ 2 moves, crossing-removal moves, and deletion of simple loops, without increasing the number of triple points.
  • Minimality has an intrinsic characterization: a connected diagram is minimal exactly when it contains no self-intersecting monogons and no parallel bigons.
  • The number of triple crossings in a standard minimal diagram is determined by the number of pairs of linked strands that are oriented in parallel.

The paper also identifies algebraic and topological structures associated with these diagrams. Local moves induce cluster-algebra exchange relations of the form f = (ac + bd)/e, connecting the construction to the multidimensional octahedron recurrence and to parametrizations of totally positive Grassmannian cells. Further connections are made with Legendrian-knot invariants and with planar algebras generated by a 3-box. Under the stated planar-algebra hypotheses, the reduction results imply that the relevant spaces are spanned by at most n! minimal diagrams, one for each admissible strand connectivity.

Influence on later research

Later work directly developed the paper’s proposed relations between relations. Balitskiy and Wellman proved that the fundamental group of the relevant plabic-graph flip graph is generated by cycles of lengths 4, 5, and 10, and used this result to prove a related conjecture of Thurston concerning triple-crossing diagrams. (arxiv.org)

The paper’s planar-algebra direction was pursued by Jones, Liu, and Ren, who classified subfactor planar algebras generated by a nontrivial 3-box satisfying a relation proposed by Thurston. Their classification includes the E^6 example and planar algebras arising from quantum SU(N) representations. (arxiv.org)

The work was published in the 2017 conference proceedings volume, while its original arXiv version dates to 2004. (arxiv.org)