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

The Mayer--Vietoris sequence for the splice identifies the companion meridian with the pattern longitude and kills exactly the relation already killed when passing from $X_{U,P}$ to $X_P$. It therefore gives a canonical isomorphismApply multiplicativity of Turaev torsion to the torus union defining $X_{CP}$, and use part (b) to replace the torsion of $X_{U,P}$. The factors $1-[{\lambda }_U]$ cancel against the peripheral factor from the companion exterior. Usingon the three knot exteriors leavesUnder a one-variable identification, ${\iota }_{\ast }^C$ sends the companion variable to $t^w$, where $w=[U]\in H_1(X_P)\cong \mathbb{Z}$ is the winding number of a satellite pattern. Thus this is also the usual Satellite formula for the Alexander polynomial.