For a compact surface 
that is both a connected space
and an orientable surface and has negative
Euler characteristic, a pants decomposition
is a collection of pairwise disjoint, pairwise nonisotopic essential simple closed
curves in the interior of 
such that cutting 
along every curve produces a disjoint
union of pairs of pants. Here essential means
that no curve bounds a disk or is isotopic
to a component of the boundary
(Buser 1992).
If 
has genus 
and 
boundary components, every pants
decomposition has 
cutting curves and produces 
pairs of pants. Thus a once-punctured torus
is cut by one curve into one pair of pants, while a double
torus is cut by three curves into two pairs of pants.
When a closed surface is given a hyperbolic geometry, the geodesic lengths of the cutting curves determine the geometry of the resulting pairs of pants, while twist parameters record how adjacent pairs of pants are reglued. Together these parameters give Fenchel-Nielsen coordinates on its Teichmüller space (Buser 1992).
