Fenchel–Nielsen twist ($\tau$)

After cutting a hyperbolic surface along a simple closed geodesic, reglue its two boundary copies after an arclength displacement $\tau$. The boundary length remains fixed, while the Riemannian metric varies. In marked Fenchel–Nielsen coordinates, the twist is real-valued; changing it by one boundary length corresponds to a Dehn twist in the marking. This construction also applies to a hyperbolic pair of pants with a fixed cone angle and two geodesic boundaries.