Standard-sphere realizations for the remaining prism-family parameters

Determine whether, for every prism-family parameter d outside the values already realized, the corresponding prism-manifold knot group admits a smooth fibred two-knot realization in the standard four-sphere.

Background

The paper places the order-seven case in the classical family of groups whose commutator subgroup is of the form Q(8)×(Z/nZ). Earlier constructions realized only a finite list of parameters, while the general existence problem remained unresolved. The present paper resolves the particular omitted parameter n=d=7, but does not resolve the corresponding question for all remaining parameters.

References

His subsequent paper enlarged the realized list to $d=3,5,11,13,19,21,27$ and again stated that such a fibred $2$-knot in $S4$ was unknown for every other value of $d$ p.~179.

— Fibred realizations of the prism quandle $P_7$ in the standard four-sphere  (2609.29839 - Jablonowski, 24 Sep 2026) in Section 1, subsection “The main result and its motivation” (historical discussion of the prism-family realization problem)