Exercise 17, § 5

We identify $\mathbf{S}_1$ with the set of complex numbers of modulus 1, and the sphere $\mathbf{S}_3$ with the subspace of $\mathbf{C}^2$ formed by the points $(z, w)$ such that $|z|^2+|w|^2= 1$. Let B and C be the subspaces of $\mathbf{S}_3$ defined by $|z|\leqslant |w|$ and $|z|\geqslant |w|$ respectively.

(a) Prove that B and C are homeomorphic to $\mathbf{B}_2\times \mathbf{S}_1$.

(b) For every pair $(m, n)$ of relatively prime integers, let $K_{m,n}$ denote the image of the mapping of $\mathbf{S}_1$ into $\mathbf{S}_3$ given by $u\mapsto (u^m, u^n)$ (“toric knot”).

(c) Prove that $\mathbf{S}_3-K_{m,n}$ is connected and that its fundamental group admits a presentation $\langle x, y;x^m=y^n\rangle$.

(d) Prove that there exists a homeomorphism $\varphi$ of $\mathbf{S}_3$ onto itself such that $\varphi (K_{m,n}) = K_{1,0}$ if and only if $|m|\leqslant 1$ or $|n|\leqslant 1$.