Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 141 / 4 / b / Solution

The manifold $X_P$ is obtained from $X_{U,P}$ by filling the $U$-boundary along ${\mu }_U$. The filling core represents ${\iota }_{\ast }^{\partial X_U}[{\lambda }_U]$ in $H_1(X_P)$. The Turaev-torsion Dehn-filling formula therefore givesSince a knot exterior has first Betti number one, its normalized Turaev torsion isSubstitution proves