= Solution
Orient the two components coherently through the twist region and assign meridian variables $x,y$. The <link diagram> is the <torus link> $T_{2,2n}$. It is obtained from the three-component <torus link> of part 2 by $-1/(n-1)$ <rational Dehn surgery> on the third component: removing its meridional disk adds $n-1$ full twists to the original single full twist. For $n=1$, this just means meridionally deleting the third component.
Write $z$ for the removed component's meridian. Its longitude is homologous to $\mu_1+\mu_2$, so the filling imposes $z=(xy)^{n-1}$. Under this substitution, the polynomial of part 2 becomes $(xy)^n-1$. The filling core is homologous, up to sign, to $\mu_1+\mu_2$. The <Turaev-torsion Dehn-filling formula> therefore removes the factor $xy-1$, giving
$$
\boxed{\Delta_{L(n)}(x,y)\doteq\frac{(xy)^n-1}{xy-1}.}
$$
For positive $n$ this is the <Laurent polynomial> $1+xy+\cdots+(xy)^{n-1}$. For $n=1$ it is one, as for a <Hopf link>; for $n=0$ it is zero, as for the two-component <unlink>. For negative $n$ the displayed quotient is still a <Laurent polynomial> and agrees with the mirrored positive-twist answer up to a unit. These checks also fix the twist count: the exponent is $n$, rather than $n+1$.
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