Q:

Knot symmetries and the Alexander polynomial

Let $K\subset S^3$ be a knot. Suppose there is an involution, $f$, of $S^3$ such that $f(K)=K$, and the fixed points of $f$ do not lie on $K$ itself. Furthermore assume that the orientations of $f(K)$ and $K$ match. For example, knots which are the closures of squared braids $\sigma^2$ have this property. I recall reading that the Alexander polynomial of such knots has some restrictions on it, but I can't seem to locate the reference where I originally read this. Any help tracking down this reference is appreciated.

A:

Murasugi
http://www.ams.org/mathscinet-getitem?mr=292060
see also
http://www.ams.org/mathscinet-getitem?mr=1133872

