A Characterization of the Murasugi Polynomial of an Equivariant Slice Knot

We characterize the Murasugi polynomial of an equivariant slice knot by proving a conjecture of J. Davis and S. Naik. A knot K in S is called periodic of period p if there is an orientation preserving action of Z/p on S which preserves K setwise and its fixed point set A is a circle disjoint to K. A is called the axis. Two periodic knots K0 and K1 of period p are called equivariantly concordant if there is an action of Z/p on S × [0, 1] such that Ki × i is periodic via its restriction on S× i, i = 0, 1, and there is a locally flat submanifold in S× [0, 1] which is preserved by the action, homeomorphic to S×[0, 1], and bounded by (K0×0)∪−(K1×1). The unknotted circle S ×0 ⊂ ∂(D ×D) = S can be viewed as a periodic knot via the (2π/p)-rotation on the first D factor. If a periodic knot K is equivariantly concordant to it, K is called an equivariant slice knot. There are several known obstructions for a periodic knot K to being an equivariant slice knot. Some of them are obtained from invariants of K. In [7], Naik used the Alexander polynomial and metabolizers of the Seifert form of K. She also showed that certain Casson-Gordon invariants of K must vanish if K is equivariant slice. In [3], Choi, Ko, and Song defined an obstruction from a Seifert matrix of K. Further obstructions are obtained by considering the quotient link. Given a periodic knot K with axis A, the orbit space of the (Z/p)-action is again S by the Smith conjecture, and the images Ā and K̄ of A and K under the quotient map form a twocomponent link which is called the quotient link. It contains all the essential information on the periodic knot. In [2], Ko and the author developed an obstruction for K to being an equivariant slice knot from knots obtained by surgery on the quotient link. In particular, their Casson-Gordon torsion invariant was used to construct an example of a non-equivariant-slice knot which cannot be detected by other invariants. Recently, in [4], Davis and Naik have studied the Murasugi polynomial ∆Z/p(g, t) of a periodic knot K, which is the image of the Alexander polynomial of the quotient link under the projection Z[Z × Z] → Z[Z/p × Z]. Here g and t are generators of Z/p and Z corresponding to the components Ā and K̄, respectively. They proved the following realization theorem of the Murasugi polynomial of an equivariant slice knot: Theorem 1 (Davis-Naik). For any a(g, t) ∈ Z[Z/p × Z] such that a(g, 1) = 1, there is an equivariant slice knot K with Murasugi polynomial ∆Z/p(g, t) = a(g, t)a(g , t). In fact, their knot K is an equivariant ribbon knot, which is a specialization of an equivariant slice knot. 2000 Mathematics Subject Classification. Primary 57M25.