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

{c}

If filling a boundary torus along a slope produces $Y$ and the filling core represents $h\in H_1(Y)$, then, up to the standard torsion unit,
$$
\tau(Y)\doteq\frac{i_*\tau(X)}{1-h}.
$$
Equivalently, $i_*\tau(X)\doteq(1-h)\tau(Y)$.


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