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Let S be a path-connected, locally-compact CW-complex, and let M be asubcomplex with finitely-many components. A `decorated SL_2(C)-local system' isan SL_2(C)-local system on S, together with a choice of `decoration' at eachcomponent 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 functionson the space of decorated SL_2(C)-local systems on (S,M) which are regular withrespect to the monodromy. The character algebra is presented explicitly. Thecharacter algebra is then shown to correspond to the algebra spanned bycollections of oriented curves in S modulo simple graphical rules.  As an intermediate step, we obtain an invariant-theory result of independentinterest: a presentation of the algebra of SL_2(C)-invariant functions onEnd(V)^m + V^n, where V is the tautological representation of SL_2(C).

Character algebras of decorated SL_2(C)-local systems