Séminaire Lotharingien de Combinatoire, 95B.88 (2026), 12 pp.
Saskia Solotko, Katherine Tung, Mengyuan Yang and Yuchong Zhang
Multitriangulations on Ciliated Surfaces
Abstract.
We generalize the definition of k-triangulations to arbitrary
ciliated surfaces, and prove that the simplicial complex
ΔCn,2 is pure and a weak
pseudomanifold of dimension 2(n-1), where
ΔCn,2 is the simplicial
complex associated with 2-triangulations on the half-cylinder, which
is an annulus with n marked points on one boundary. This result
generalizes the work of Vincent Pilaud and Francisco Santos for
polygons and resolves a conjecture of Mathias Lepoutre and Vincent
Pilaud for k=2. To achieve this, we show that 2-triangulations
on the half-cylinder decompose as complexes of star polygons, and that
2-triangulations on the half-cylinder are in bijection with
2-triangulations on the 4n-gon invariant under rotation by
π/2 radians. Building on work by Vincent Pilaud and Christian
Stump, we also introduce chevron pipe dreams, a new combinatorial
model that more naturally captures the symmetries
of k-triangulations.
In addition, we propose some conjectures on k-triangulations of
general ciliated surfaces.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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