  Given a two-dimensional substitution tiling space, we show that, under some
reasonable assumptions, the $K$-theory of the groupoid $C^\ast$-algebra of its
unstable groupoid can be explicitly reconstructed from the $K$-theory of the
$AF$-algebras of the substitution rule and its analogue on the $1$-skeleton. We
prove this by generalizing the calculations done for the chair tiling in [JS16]
using relative $K$-theory and excision, and packaging the result into an exact
sequence purely in topology. From this exact sequence, it appears that one
cannot use only ordinary $K$-theory to compute using the dimension-filtration
on the unstable groupoid. Several examples are computed using Sage and the
results are compiled in a table.
