Cyclic group actions on odd-dimensional spheres

We show that for any simple (2q−1)-knotk, q>1, and any positive integern, the knot # ₁ ⁿ k is the fixed-point set of aZ _n -action onS ^2q+1. Further, we show that for many values ofn there are examples of (2q−1)-knots,q≥2, which are the fixed-point sets of inequivalentZ _n -actions. This paper was written whilst the first author was in receipt of a Research Grant from the Science Research Council of Great Britain.