Let $K$ be a knot whose Alexander polynomial is not trivial and $G = \pi_1(S^3K)$. By Tim Cochran's noncommutative theory, $G$ is not solvable. But is $G$ a hypoabelian group? In particular, I'm interested in twisted Whitehead doubles of trefoil knot $K'$. Is it possible that the transfinite derived series of $\pi_1(S^3  K')$ terminates at a nontrivial perfect core at stage $\omega$, where $\omega$ is a limit ordinal? A related question is, can $G_H^{(\omega)}$ be trivial, where $G_H^{(\omega)}$ is the torsionfree derived series developed by Harvey?
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$\begingroup$ The first part of the above question is negative. It has been answered by Ian Agol in a related question link $\endgroup$ – Shijie Gu Sep 14 '17 at 7:09