I have the following problem that I cannot solve:
Let $T\subset S^3$ be the standard torus and $T(p,q)$ be a simple closed curve lying on $T$, representing the $pr+qs$ homology class. Let $Y=S^3-\nu(T(p,q))$. Use Seifert-Van-Kampen to show that $\pi_1(Y)=\langle x,y \mid x^p=y^q \rangle$.
I believe that $T(p,q)$ is what is usually called a torus knot. I have seen in Hatcher how to find the fundamental group, but there the torus knot is defined as the embedding of $S^1 \rightarrow S^1\times S^1$ via tha map $z\mapsto (z^p,z^q)$. Moreover, the proof there is quite lengthy. Is there an easy way to compute the fundamental group given this definition of $T(p,q)$?
Best Answer
Hatcher's explanation is to demonstrate how to deformation retract $Y$ onto a CW complex, since it is a useful technique for calculating homotopy and homology groups.
A sketch of a more direct use of the van Kampen theorem: We may decompose $S^3-T$ into two open solid tori $A,B$. Let $C$ be $T-T(p,q)$ thicked up a little. Then, $A\cup C$ and $B\cup C$ form an open cover of $Y$ with path connected intersection. $\pi_1(A\cup C)\cong\pi_1(A)$ is generated by some loop $x$ and $\pi_1(B\cup C)\cong\pi_1(B)$ is generated by some loop $y$. The intersection $C$ is a thicked annulus, so it too is generated by some loop $z$. Including $z$ into $A\cup C$ or $B\cup C$ gives $x^p$ or $y^q$, respectively (perhaps swapping $p$ and $q$ depending on the convention). The van Kampen theorem says that $\pi_1(Y)\cong\langle x,y\mid x^p=y^q\rangle$, since this is $\pi_1(A\cup C)*_{\pi_1(C)}\pi_1(B\cup C)$.