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arXiv · 2604.26354

On enumeration of $b$-angulations of surfaces from an integrability perspective

Elba Garcia-Failde·Jianghao Xu·Di Yang·Don Zagier
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Abstract

In this paper, we study generating series enumerating polygonal angulations of closed oriented surfaces of fixed genus, focusing on $b$-angulations with $b = 3$ or $b = 2\nu$, $\nu \geq 2$. Based on Toda integrability, we establish new structural results in the cases $b = 3$ and $b = 4$. Furthermore, via the Hodge--GUE correspondence, we derive a fine structure in the $b = 2\nu$ case, which implies a conjectural statement of Gharakhloo--Latimer.

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Elba Garcia-Failde, Jianghao Xu, Di Yang, Don Zagier. 2026-04-29. On enumeration of $b$-angulations of surfaces from an integrability perspective. https://arxiv.org/abs/2604.26354

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