Higher-dimensional analogue of Pao’s branched cover theorem for twist-spun knots

Establish whether, for n ≥ 2, the m-fold cyclic branched cover of S^{n+3} along a k-twist-spun (n+1)-knot τ_k(K^n) is diffeomorphic to S^{n+3}, with branch locus equal to a branched twist spin of K^n, generalizing Pao’s theorem from dimension 4.

Background

Pao proved that in dimension 4 the cyclic branched cover of S^4 along a twist-spun knot is again S^4, and its branch set is a branched twist spin. This result is a key ingredient in the proof of Theorem 1.1 (triviality of τ{m2}(τ{m1}(K)) for classical knots when gcd(m1, m2) = 1).

The authors explain that a similar statement is not known in higher dimensions, and the lack of such a theorem prevents extending their 4-dimensional triviality result to higher-dimensional settings.