Let $T=S^{1} \times S^{1}, U=S^{1} \times D^{2}$ and $V=D^{2} \times S^{1}$. Let $i: T \rightarrow U, j: T \rightarrow V$ be the natural inclusion maps. Consider the space $S:=U \cup_{T} V$; that is,

$$S:=(U \sqcup V) / \sim$$

where $\sim$ is the smallest equivalence relation such that $i(x) \sim j(x)$ for all $x \in T$.

(a) Prove that $S$ is homeomorphic to the 3 -sphere $S^{3}$.

[Hint: It may help to think of $S^{3}$ as contained in $\mathbb{C}^{2}$.]

(b) Identify $T$ as a quotient of the square $I \times I$ in the usual way. Let $K$ be the circle in $T$ given by the equation $y=\frac{2}{3} x \bmod 1 . K$ is illustrated in the figure below.

Compute a presentation for $\pi_{1}(S-K)$, where $S-K$ is the complement of $K$ in $S$, and deduce that $\pi_{1}(S-K)$ is non-abelian.