An invariant knot is a triple \((J^{2n+1},K^{2n-1};T)\) where \((J,K)\) is an oriented simple knot (smooth or PL), \(n\geq 2\), and \(T\) is a free \(\mathbb Z_ m\)-action on the ambient sphere \(J\) under which \(K\) is invariant. Such knots \((J,K)\) were classified by J. Levine in terms of the Seifert matrix modulo \(S\)-equivalence, and here such invariant knots are classified in terms of the quotient normal bundle \(\nu (K/T,J/T)\) and something called the derived Seifert pairing \(\beta\). It turns out that the quotient under \(T\) of the knot exterior \(X\) is also a knot exterior \(X^*\), and \(\beta\) is the Seifert pairing associated with \(X^*\). It is shown that \(\nu\) and \(\beta\) determine the triple \((J,K;T)\) up to equivariant homeomorphism.
The author also characterises the matrices which arise as Seifert matrices of the knot \((J,K)\) where \((J,K;T)\) is an invariant knot. Since \(X^*\) is a knot exterior, it follows that \((J,K)\) is in fact a branched cyclic cover of a simple knot, and so this work is related to that of P. M. Strickland [Proc. Am. Math. Soc. 90, 440–449 (1984; Zbl 0543.57012)].