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A knot is understood as an oriented 1-sphere piecewise linearly imbedded in an oriented 3-sphere unless otherwise stated. A knot k C S 3 is amphicheiral if S 3 admits an orientation-reversing piecewise-linear autohomeomorphism h such that h(k)=k. In particular, if S 3 admits an orientation-reversing piecewise-linear involution h (i.e. h2=id) such that h(k)=k, then we say that the knot kCS 3 is strongly amphicheiral. 4] and Kirby [-10], Problem 1.5.) More precisely, we say that the knot k C S 3 is (strongly) positive or negative amphicheiral according to whether the restricted autohomeomorphism (involution) h[k :k-~k is orientation-preserving or-reversing.Let k C S 3 be a knot. Let ( t ) be a fixed infinite cyclic group generated by a letter t. Let ~ :7z1(S 3 -k )~( t ) be the epimorphism determined by mapping the generator of Ht(S 3 -k ; Z ) which has linking number + 1 with the knot k to t. Let be the cover S 3 -k associated with the epimorphism ),. We denote by H(k) the rational homology group HI(SS-Z-k-k;Q). It is well-known that H(k) is a finitely generated torsion Q(t)-module, called a knot module (over Q(t)). We call any polynomial A(t) with IA(1)I=I generating the Q(t)-order ideal of H(k) the Alexander polynomial of the knot k. Note that A(t) becomes a Laurent polynomial with integer coefficients. (Cf. I-7], Lemmas 2.6 and 2.7.)We shall show the following: tTheorem.(1) The Alexander polynomial A(t) of a strongly negative amphicheiral knot sati,~fies the equality A(t2) -

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