Higher-dimensional triviality of iterated twist-spinning under coprime twists

Determine whether, for n ≥ 2 and integers m1, m2 ≥ 1 with gcd(m1, m2) = 1, the iterated twist-spinning τ_{m2}(τ_{m1}(K^n)) of an n-dimensional knot K^n ⊂ S^{n+2} is the trivial (n+2)-knot in S^{n+4}, extending Theorem 1.1 beyond the classical (n = 1) case.

Background

Zeeman’s k-twist-spinning constructs an (n+1)-dimensional knot in S^{n+3} from an n-dimensional knot in S^{n+2}. The paper proves that for classical knots (n = 1), if gcd(m1, m2) = 1 then the m2-twist-spinning of the m1-twist-spinning is a trivial 3-knot in S^5 (Theorem 1.1).

The authors note that even when τ{m1}(K) is non-trivial, τ{m2}(τ_{m1}(K)) becomes trivial if gcd(m1, m2) = 1, and raise the question of whether an analogous triviality statement persists in higher dimensions (n ≥ 2). Remark 3.4 further explains that the current proof relies on Pao’s 4-dimensional branched cover theorem, for which higher-dimensional analogues are not known.