On Equivariant Slice Knots

We suggest a method to detect that two periodic knots are not equivariantly concordant, using surgery on factor links. We construct examples which satisfy all known necessary conditions for equivariant slice knots — Naik’s and Choi-Ko-Song’s improvements of classical results on Seifert forms and Casson-Gordon invariants of slice knots — but are not equivariantly slice. Introduction An oriented knot K in S is called a periodic knot of period p, or a p-periodic knot if there is an orientation preserving action of the cyclic group Zp on S 3 whose fixed point set is a circle (called an axis) disjoint to K, and K is invariant under the action. By the confirmed Smith conjecture, we may assume the cyclic group Zp is generated by the standard (2π/p)-rotation around an unknotted axis. It is not known whether a knot can be invariant under more than one Zp-actions. In this paper we regard a p-periodic knot as a knot equipped with the Zp-action around a fixed axis, denoted by A. Two knots K0 and K1 are called concordant if there is a proper submanifold C in S×[0, 1] diffeomorphic to S×[0, 1] such that C∩(S×0) = K0, C∩(S ×1) = −K1. C is called a concordance between K0 and K1. Two p-periodic knots K0 and K1 are equivariantly concordant if there is a concordance C which is invariant under an extended Zp-action on S 3 × [0, 1] that extends the standard Zp-actions on S 3 × 0 and on S×1 with the fixed axis. An extended Zp-action on S × [0, 1] is sometimes required to be standard so that it fixes A× [0, 1] [4]. Since there is a Zp-action on S that fixes a knotted S [6], this restriction seemingly gives a stronger relation than the equivariant concordance defined here. A knot is called a slice knot if it is concordant to the unknot. The unknot that forms a hopf link with the axis A can be considered as a p-periodic knot in an obvious way, and a periodic knot is called equivariantly slice if it is equivariantly concordant to the unknot. In the case of knots without periodicity, many invariants under knot concordance are known. One of the most important invariants is the Seifert form. In [11], J. Levine showed that the Seifert form of a slice knot admits a metabolizer. In the case of odd higher dimensional knots, he also showed that the converse is true. However, in the case of 1-dimensional knots, a secondary obstruction is found by 1991 Mathematics Subject Classification. Primary 57M25, 57M60; Secondary 57Q60.