For each natural number $n$ let $Z_n$ be the $n$-clover obtained by forming the disjoint union of $n$ circles, choosing one point in each circle, and then identifying these $n$ points, so that they become one common point. In particular, $Z_1=\Bbb T$ and $Z_2$ is the “figure eight”.Calculate $K_0(C(Z_n))$ and $K_1(C(Z_n))$ for each $n$ by constructing a short exact sequence $0→C_0(U)→C(Z_n)→C(Y)→0$ for suitable spaces $U$ and $Y$.
Here,
for $K_0(C(Z_2))$, can i use the fact that $K_0$ is isomorphic to the free abelian group generated by isomorphism classes of finitely generated projective modules over the given $C^*$-algebra. In this case, $C(Z_2)$ is the $C^*$-algebra associated with the figure-eight space.
and for $K_1(C(Z_n))$ , the fact that $K_1(C(X))$ is isomorphic to the stable homotopy group $\pi_1(X)$, the fundamental group of the space $X$. For $Z_1 = T$ and $Z_2 $= the figure-eight space.
Is there any other alternative method? provide some hints