Fenchel–Nielsen coordinates ($({\ell }_1,\dots ,{\ell }_{3g-3},{\tau }_1,\dots ,{\tau }_{3g-3})$)

For a fixed pants decomposition, the positive cuff lengths and real twist parameters give coordinates on Teichmüller space. With twists measured in length units, a full Dehn twist changes ${\tau }_i$ by ${\ell }_i$; with angle units it changes the twist by $2\pi$.

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.

For a fixed pants decomposition of a closed genus-$g$ hyperbolic surface, its $3g-3$ cuff lengths determine all its Y-pieces. Two suitably chosen transverse simple closed curves at each cuff, one obtained from the other by a Dehn twist, determine the corresponding twist: hyperbolic seam identities express their lengths through shifted hyperbolic cosines of that twist, removing the sign ambiguity of just one transversal. Thus finitely many marked lengths determine the Fenchel–Nielsen coordinates. The unmarked length spectrum lacks these labels, which is why this fact alone does not imply global spectral uniqueness.