Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 141 / 2 / b / Solution

The curve $\mu$ crosses the alpha disk ${\alpha }_2$ once and is disjoint from ${\alpha }_1$, so in the presentation it represents $y$ up to inversion and conjugacy. The Dehn filling adds the relation $y=1$, after which the beta relator becomes $x^2$. Thus the filled genus-one Heegaard diagram has two alpha–beta intersections andCompressing the displayed genus-two diagram along $\mu$ visibly cancels the annular handle and leaves that genus-one Heegaard diagram. It is therefore the lens space