STABILIZING HEEGAARD SPLITTINGS OF COMPOSITE TUNNEL NUMBER TWO KNOTS

Abstract

Suppose K # K′ is a composite tunnel number two knot where both K and K′ are tunnel number one knots. Let t1, t2 be unknotting tunnels of K and t′1 , t′2 be unknotting tunnels of K′ and assume that cl (S3 - N(K ∪ t1 ∪ t2)) and cl (S3 - N(K′ ∪t′1 ∪t′2)) are genus three handlebodies. Suppose {t1 , t′1} , {t2 , t′2} are tunnel systems of K # K′. Then we give some condition such that the two genus three Heegaard splittings induced by these tunnel systems become isotopic by one stabilization. On the other hand, we construct infinitely many examples such that cl (S3 - N(K # K′ ∪ t1 ∪ t′1 ∪ t′2)) is not a genus four handlebody and give candidates of two splittings which cannot be made isotopic by a single stabilization.