A note on a Heegaard diagram of S^3

Ochiai describes a Heegaard diagram of S^3, with a particular associated fundamental group presentation H.This Heegaard diagram has neither waves nor pairsof complementary handles, so it is directly reducible neither by awave-move , nor by a Singer's move of type III'. Hence it is a counterexample to theWhitehead's conjecture and to the algorithm A ofVolodin-Kuznetsov-Fomenko.We construct a crystallization of S^3 having H as an associated presentation of the fundamentalgroup with respect to a suitable choice of generators and relatorsand such that Ochiai's Heegaard diagram is one among the Heegaard diagramsassociated to the crystallization.Moreover, we prove that at least one among the remaining Heegaarddiagrams associated to our crystallization hassome pairs of complementary handles, and so it is directly reducibleby Singer's move of type III' to the canonical diagram ofS^3.