Conway polynomials of periodic links

Abstract

Introduction Geometric properties of knots and links in a 3-sρhere *S often have effect on their polynomial invariants. Periodicity of knots and links is one of them. Therefore to study periodic knots and links, it is significant to investigate their polynomial invariants. In this paper, we consider the following situation: Let L=Kι\J ••• UKμy μ> be an oriented link and B be a trivial knot with Bf]L=0. We consider the />-fold cyclic cover qp: S ->S branched over B, wherep>2. We denote the preimage of L and K{ by L and Kiy respectively and call them the covering links of L and Kh respectively. Let Ki=Kn U ••• U Kiv. be a -component link. We give K(j the orientation inherited from K{. Then L=Kλ U U J?μ=ϋΓiί U U Kλ Vl U U Kμi U U Kμ*μ has the unique orientation. In 1971, Murasugi [7] showed a relationship between the Alexander polynomials of L and L for the case μvμ=l. Later, Hillman [4], Sakuma [10] and Turaev [12] extended Murasugi's result to the general case. Our goal in this paper is to give a relation between the Conway polynomials of L and L using their results (Theorem 2). To do this, we sharpen their formulas by expressing in terms of the Conway potential function [2] whose existence is shown by Hartley [3] (Theorem 1). Although the Alexander polynomial is usually defined with the difference by a unit of a polynomial ring, the Conway potential function is uniquely defined as an element of a polynomial ring. We denote the Conway potential functions of L, B U L and Lby ΩL(tι> •••, tμ), ΩB[iL(s, tu—9 U) and Ωz(tn, •••, /1Vl, •••, tμi, —,'f<ιV), respectively, where tiy s and tjj correspond to Ki} B and Kijy respectively. Then concerning the Conway potential functions of L and L, the following formula holds: