  This paper deals with a complete invariant $R_X$ for cyclic quotient surface
singularities. This invariant appears in the Riemann Roch and Numerical
Adjunction Formulas for normal surface singularities. Our goal is to give an
explicit formula for $R_X$ based on the numerical information of $X$, that is,
$d$ and $q$ as in $X=X(d;1,q)$. In the process, the space of curvettes and
generic curves is explicitly described. We also define and describe other
invariants of curves in $X$ such as the LR-logarithmic eigenmodules,
$\delta$-invariants, and their Milnor and Newton numbers.
