The manifold $X_P$ is obtained from $X_{U,P}$ by filling the $U$-boundary along $\mu_U$. The filling core represents $\iota_*^{\partial X_U}[\lambda_U]$ in $H_1(X_P)$. The <Turaev-torsion Dehn-filling formula> therefore gives
$$
\iota_*^{X_{U,P}}\tau(X_{U,P})
=\bigl(1-\iota_*^{\partial X_U}[\lambda_U]\bigr)\tau(X_P).
$$
Since a knot exterior has first Betti number one, its normalized <Turaev torsion> is
$$
\tau(X_P)=\frac{\Delta(X_P)}{1-\iota_*^{\partial X_P}[\mu_P]}.
$$
Substitution proves
$$
\boxed{
\iota_*^{X_{U,P}}\tau(X_{U,P})
=\frac{\Delta(X_P)\bigl(1-\iota_*^{\partial X_U}[\lambda_U]\bigr)}
{1-\iota_*^{\partial X_P}[\mu_P]}.}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-141/4/b/solution