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By J.C. Cantrell

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L, 1 1, < i < n. -- Then is a Heegaa:rad sewing associated to the IT T )( IT T T ) · 1 ftt... J= · 1 x. 2. = J J cLass 1 diagpam D. Proof. 2 the curve systems x, + + f, z satisfy the Ix . 1 ;:: J J intersection properties 1, <5... 4) T x .. 1- T 2 .. -z, (x ) ::::: x k := (4 .. (Zk) ;::: zk if k Fi ;::: f 1,.. if k := t. ) ;::: f··, 1, 1 ~ i ~ n, i as required. 8. 7 the map h is expressed as a product of Dehn twists, hence we may apply the method of Lickorish [4J to produce a framed link which defines M(h).

Is a U f. c aA. cylinder, it follows that aA split open along f is a punctured ~ n sphere, hence f is a complete system for aA • n ~ Next we choose a second system of curves x = {x ' ••• ,x } c aA ~ 1 n n which bounds a complete system for A. The curves x will be select so that each x. encircles theni th handle once in the manner 1, ~~ Let D = (aA ,x" f) • Then D is a Heegaard n diagram for a 3-manifold M(V}. It is the diagpam associated to indicated in Figure 4. 1 • the cZass-l fPamed Zink L (with connecting edges e).

Of Math. 59, No. 2, 1975, pp. 475486. 8. , "A note on branched cyclic coverings of spheres", Rev. Mat. -Amer. 32 (1972), pp, 158-166. 9. Suzuki, Shinicni, "On homeomphisms of a 3-dimensional handlebody", Can. J. Math. XXXIX, No. 1, 1977, pp. 111-124. 10. , "Constructing framed 4-manifolds with given almost-framed boundary*', preprint. 11. Milnor, J. , Symmetric bilinear forms, Ergibnisse der mathematik, Band 73, Springer-Verlag, 1973. 12. Powell, Jerome. "Homeomorphisms of the 3-sphere that leave a Heegaard handlebody invariant", to appear.

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