{c}
{title2=$X(\ell_1,\ell_2,\ell_3,\ell_4;s,\tau)$}

= X-piece
{c}
{synonym}

An <X-piece> is a curvature-$-1$ four-holed sphere with geodesic boundary, formed by gluing two <Y-pieces> along equal-length cuffs. For a specified separating cuff and labelled outer boundaries, four outer lengths, the internal cuff length $s>0$, and a twist $\tau$ determine the metric. The gluing is orientation-reversing on boundary circles, with relative arclength displacement $\tau$ measured from chosen seam endpoints. Without markings, the gluing displacement is taken modulo $s$; with a marking, $\tau\in\mathbb R$ records <Dehn twists> as well. Changing the chosen separating curve can give different parameter descriptions of the same unmarked surface.


 Hyperbolic X-piece (source code)