The three-sphere \[ S^3=\{(z,w)\in\mathbb C^2\mid |z|^2+|w|^2=1\} \] is the union of \[ H_1=\{(z,w)\in S^3\mid |z|^2\leq \tfrac12\} \quad\text{and}\quad H_2=\{(z,w)\in S^3\mid |w|^2\leq \tfrac12\}. \] Check that each piece is a solid torus, homeomorphic to \(B^2\times S^1\), and that their intersection is a torus \(T\).
Let \(K\) be a torus knot of type \((p,q)\) in \(T\), where \(p\) and \(q\) are coprime positive integers. Thus, \(K\) travels \(p\) times longitudinally and \(q\) times meridionally. By partitioning \(S^3-K\) with \(H_1-K\) and \(H_2-K\), calculate a presentation for \(\pi_1(S^3-K)\).
Solution
1. Each \(H_i\) is a solid torus, and \(H_1\cap H_2\) is the torus\[ T=\{(z,w)\in S^3:|z|=|w|=1/\sqrt2\}. \]
For \(H_1\), write \(z\in D^2_{1/\sqrt2}\) freely and write \[ w=\sqrt{1-|z|^2}\,u,\qquad u\in S^1. \] This gives \(H_1\cong D^2\times S^1\). The analogous parametrization gives \(H_2\cong S^1\times D^2\). Their intersection is characterized by equality in both inequalities, hence \(|z|=|w|=1/\sqrt2\), a product of two circles.
2. Put \(U_i=H_i\setminus K\). Each \(U_i\) deformation-retracts to the core circle of the solid torus, so\[ \pi_1(U_1)=\langle x\rangle\cong\mathbb Z,\qquad \pi_1(U_2)=\langle y\rangle\cong\mathbb Z. \]
The knot \(K\) lies on the boundary torus. Push the boundary slightly inward away from \(K\) and then use the ordinary radial deformation retraction of a solid torus onto its core circle. This gives a deformation retraction of each \(H_i\setminus K\) onto its core.
3. The intersection \(U_1\cap U_2=T\setminus K\) is an open annulus, hence has fundamental group \(\mathbb Z\), generated by a loop parallel to \(K\).
Removing an essential simple closed curve from a torus leaves a cylinder. A core circle of this cylinder is isotopic in \(T\) to \(K\).
4. Under the inclusions into \(U_1\) and \(U_2\), this generator maps to \(x^p\) and \(y^q\), respectively.
By hypothesis the \((p,q)\) torus knot travels \(p\) times longitudinally in the first solid torus and \(q\) times longitudinally in the complementary solid torus (equivalently, meridian and longitude are interchanged between the two Heegaard solid tori). The meridional component bounds a disk in the corresponding solid torus and hence contributes trivially to its fundamental group. Thus the class of \(K\) becomes \(x^p\) in \(U_1\) and \(y^q\) in \(U_2\).
5. Seifert–van Kampen therefore gives\[ \boxed{\pi_1(S^3\setminus K)\cong\langle x,y\mid x^p=y^q\rangle.} \]
The union \(U_1\cup U_2\) is \(S^3\setminus K\). Van Kampen identifies the two images of the generator of \(\pi_1(U_1\cap U_2)\), imposing the single relation \(x^p=y^q\) on the free product \(\langle x\rangle*\langle y\rangle\).