Theorem 41.2:

Let $K$ be a subset of $S^n$ that is homeomorphic to $I^k$ for some $k$ in the range $0 \leq k \leq n$. Then

$$H_j(S^n - K) = \left\{\begin{array}{lll} \mathbb{Z} & & \mbox{ if } j = 0, \\ 0 & & \mbox{ if } j > 0. \\ \end{array} \right.$$