$H_i(S ^2 - S^1 \vee S^1) $

Question

Recently I am reading Hatcher book homology part section 2.B after a bit of reading this section I am trying the exercise 1 at page 176 , while solving this exercise I am struggling.. That's why I am asking the simple version of the exercise bellow

Let $X$ be a subspace in $S^2$ which is homeomorphic to $S^1\vee S^1$ then what is the homology groups of $S^2 \setminus S^1\vee S^1$.

Thanks in advance.

Answer 1

I hope the picture settles the idea.

The point where the two circles (of the figure eight) touch is not in $S^2\setminus S^1\vee S^1$.

This complement consist of the three gray disks.