# Pull-back of knots in branched covers and the Alexander polynomial

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 $K \subset S^3$?

Corollary 4.2 from the paper "Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation" by Herald, Kirk and Livingston gives the followwing recipe to compute Alexander polynomial of the lift of $K$ to the $n$-fold cover $\Sigma_n(K)$: $$\Delta_{\widetilde{K}_n}(t) = \prod_{i=0}^{n-1} \Delta_{K}(\xi_n^i t^{1/n}),$$ where $\xi_n = \exp(\frac{2 \pi i}{n})$.