Smooth diffeomorphism types of the cyclic threefold branched covers

Determine the smooth diffeomorphism types of the third cyclic branched covers of the prism-quandle realizations, namely $X_{+,3}$ and $X_{-,3}$, beyond establishing that they are not homotopy equivalent or stably homeomorphic.

Background

The third cyclic branched covers associated with the two framing choices have fundamental group Q8, identical basic homological data, and different Teichner secondary invariants. The paper uses that invariant to prove that the two covers are not homotopy equivalent, but it does not identify either cover up to a complete smooth diffeomorphism classification. The unresolved status is explicitly inherited from the prior monograph, which had not identified the cyclic threefold cover.

References

Theorem~21.3 of treats the double cover and gives explicit smooth models for the regular and irregular dihedral and the sixfold cyclic covers; it does not identify the smooth diffeomorphism type of the double cover, and the monograph explicitly leaves the cyclic threefold cover unidentified as well; see p.~377 and Section~21.3.

— Fibred realizations of the prism quandle $P_7$ in the standard four-sphere  (2609.29839 - Jablonowski, 24 Sep 2026) in Section “The two knots are inequivalent,” paragraph “Scope relative to the monograph”