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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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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. –  Ian Agol Jan 20 '12 at 18:08

1 Answer 1

up vote 4 down vote accepted

Murasugi

http://www.ams.org/mathscinet-getitem?mr=292060

see also

http://www.ams.org/mathscinet-getitem?mr=1133872

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Thanks, the first paper was exactly the paper I was trying to remember. –  Jim Conant Jan 20 '12 at 19:22

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