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.
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$\begingroup$ I don't know the reference, but if it's any help, one can show that there is a minimal-genus Seifert surface left invariant by the involution. Also, the involution must have fixed point set an unknot by the Smith conjecture, so your knot is a branched cover over some other knot. $\endgroup$– Ian AgolCommented Jan 20, 2012 at 18:08
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$\begingroup$ Thanks, the first paper was exactly the paper I was trying to remember. $\endgroup$ Commented Jan 20, 2012 at 19:22