Twist spun knots of twist spun knots of classical knots

A k-twist spun knot is an $n+1$-dimensional knot in the $n+3$-dimensional sphere which is obtained from an n-dimensional knot in the $n+2$-dimensional sphere by applying an operation called a k-twist-spinning. This construction was introduced by Zeeman in 1965. In this paper, we show that the m₂-twist-spinning of the m₁-twist-spinning of a classical knot is a trivial $3$-knot in S⁵ if (m₁,m₂)=1. We also give a sufficient condition for the m₂-twist-spinning of the m₁-twist-spinning of a classical knot to be non-trivial.