  Let S be a path-connected, locally-compact CW-complex, and let M be a
subcomplex with finitely-many components. A `decorated SL_2(C)-local system' is
an SL_2(C)-local system on S, together with a choice of `decoration' at each
component of M (a section of the stalk of an associated vector bundle).
  We study the (decorated SL_2(C)-)character algebra of (S,M), those functions
on the space of decorated SL_2(C)-local systems on (S,M) which are regular with
respect to the monodromy. The character algebra is presented explicitly. The
character algebra is then shown to correspond to the algebra spanned by
collections of oriented curves in S modulo simple graphical rules.
  As an intermediate step, we obtain an invariant-theory result of independent
interest: a presentation of the algebra of SL_2(C)-invariant functions on
End(V)^m + V^n, where V is the tautological representation of SL_2(C).
