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 $m_2$-twist-spinning of the $m_1$-twist-spinning of a classical knot is a trivial $3$-knot in $S^5$ if $\gcd (m_1,m_2)=1$. We also give a sufficient condition for the $m_2$-twist-spinning of the $m_1$-twist-spinning of a classical knot to be non-trivial.