Unimodality and Cluster Algebras from Surfaces
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.
Cite
@article{arxiv.2508.04396,
title = {Unimodality and Cluster Algebras from Surfaces},
author = {Wonwoo Kang and Kyeongjun Lee and Eunsung Lim},
journal= {arXiv preprint arXiv:2508.04396},
year = {2026}
}