On the additivity of $h$-genus of knots

Abstract

We say that (Vu V2; F) is a Heegaard splitting of the 3-sphere S , if both F xand V2 are handlebodies, S 3=V1ΌV2 and V1nV2=dVι=dV2=F. Then F is called a Heegaard surface of *S. Let K be a knot in S. Then it is well known that there exists a Heeagard surface of S which contains K, Thus we define h(K) as the minimum genus among all Heegaard surfaces of S containing K, and we call it the A-genus of K. We note here that any two Heegaard surfaces of *S with the same genus are mutually ambient isotopic ([11]). By the definition, it follows that h(K)=0 if and only if K is a trivial knot and that h(K)=l if and only if K is a torus knot. Hence if h(K)=l then K is prime. In this paper we show: