3 problems

Optimality conjecture. The configuration space of proper decorations is optimal if and only if all cone angles at vertices in $V_{-1}$ satisfy

Let ${\mathbf{S}}_\bigtriangleup$ be the decorated marked surface. Let $\operatorname{OA}^\circ({\mathbf{S}}_\bigtriangleup)$ be the set of open arcs appearing in triangulations in…

Let ${\mathbf{S}}_\bigtriangleup$ be the decorated marked surface, let $\operatorname{CA}({\mathbf{S}}_\bigtriangleup)$ denote its set of closed arcs, and let…