Can someone identify the knot whose complement in $\mathbb{S}^{3}$ will produce the Hantzsche-Wendt manifold?
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Sign up to join this communityCan someone identify the knot whose complement in $\mathbb{S}^{3}$ will produce the Hantzsche-Wendt manifold?
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Supposing the question is, "Can the Hantsche-Wendt manifold be realized as a cyclic branched cover of a knot in the 3-sphere", then the answer is yes: it is a 3-fold cyclic branched cover along the figure-eight knot. Incidentally, it is also a 2-fold branched cover along the Borromean rings.
On the other hand, if the question is, "Can the Hantsche-Wendt manifold $M$ be realized by performing Dehn surgery along a knot in the 3-sphere" (as Misha seems to suggest), then the answer is no, since $H_1(M,\mathbb{Z})=\mathbb{Z}_2 \oplus \mathbb{Z}_2$ is not a cyclic group.
Much enlightenment (though not an explicit answer to the question) can be gleaned from Bruno Zimmermann's paper "On the Hantzsche-Wendt manifold".