Seifert matrices of periodic knots

Abstract

We characterize the Seifert matrices of periodic knots in S-3 up to S-equivalence. Given a periodic knot we construct an equivariant spanning surface F and choose a basis for H-1(F) in such a way that the Seifert matrix has a special form exhibiting the periodicity. Conversely, given such a Seifert matrix we construct a periodic knot that realizes it. We exhibit the decomposition of H-1(F; C) into eigenspaces of the periodic action, orthogonal to each other with respect to the Seifert pairing. Consequently we obtain Murasugi's formula for the Alexander polynomial of the periodic knot.