  Under suitable conditions, a substitution tiling gives rise to a Smale space,
from which three equivalence relations can be constructed, namely the stable,
unstable, and asymptotic equivalence relations. We denote with $S$, $U$, and
$A$ their corresponding $C^*$-algebras in the sense of Renault. In this article
we show that the $K$-theories of $S$ and $U$ can be computed from the
cohomology and homology of a single cochain complex with connecting maps for
tilings of the line and of the plane. Moreover, we provide formulas to compute
the $K$-theory for these three $C^*$-algebras. Furthermore, we show that the
$K$-theory groups for tilings of dimension 1 are always torsion free. For
tilings of dimension 2, only $K_0(U)$ and $K_1(S)$ can contain torsion.
