The generators are the alpha-compressing disks, equivalently the four bounded regions. Each beta curve supplies the relator read from its signed intersections with the alpha curves. Since the basepoint region has generator $r_0=1$, the presentation is
$$
\boxed{\pi_1(X_T,x_0)
\cong\langle a,b,c,d\mid ad^{-1}b,\ bd^{-1}c,\ cd^{-1}a\rangle.}
$$
The relations say $d=ba=cb=ac$. Eliminating $c,d$ leaves $bab=aba$, the standard two-generator presentation of the <trefoil knot> group, which checks the region conventions.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 141 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-141/1/b/solution