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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 $K \subset S^3$?
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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 $\xi_n = \exp(\frac{2 \pi i}{n})$.

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  • $\begingroup$ Thank you very much, this is even better than the answer I was hoping for! $\endgroup$ – Antonio Alfieri Dec 13 '17 at 11:35

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