Given a knot $K \subset S^3$ one can form its double branched cover $\Sigma_2(K)$ and consider the pull-back knot $\widetilde{K} \subset \Sigma_2(K)$ of $K$ to $\Sigma_2(K)$ (the locus fixed by the involution of $\Sigma_2(K)$). Here a few questions I would like to ask

- How do I compute the Alexander polynomial of $\widetilde{K}$?
- Can I compute $\Delta_{\widetilde{K}}(t)$ from $\Delta_K(t)$ plus some torsion invariants of $\Sigma_2(K)$?
- Is there a formula for $\Delta_{\widetilde{K}}(t)$ in the case when $\widetilde{K}$ comes from a torus knot $K \subset S^3$?