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arXiv · 2508.04396

Unimodality and Cluster Algebras from Surfaces

Wonwoo Kang·Kyeongjun Lee·Eunsung Lim
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Abstract

We prove that the rank polynomial of the lattice of order ideals of a loop fence poset is unimodal. This poset arises as the poset of join-irreducibles in the lattice of good matchings of loop graphs associated with notched arcs. Equivalently, such polynomials can be obtained by evaluating all coefficient variables in an F-polynomial at a single variable q. We also conclude that the rank polynomial of any tagged arc, whether plain or notched, is not only unimodal but also satisfies a symmetry condition known as almost interlacing. Furthermore, when the lamination consists of a single curve, the cluster expansion-evaluated by setting all cluster variables to 1 and all coefficient variables to q-is also unimodal. We conjecture that polynomials in this case are log-concave.

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Wonwoo Kang, Kyeongjun Lee, Eunsung Lim. 2026-02-11. Unimodality and Cluster Algebras from Surfaces. https://doi.org/10.1016/j.ejc.2026.104392

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