Heegaard splitting for sutured manifolds and Murasugi sum (Q1199251)

A sutured manifold \((M,\gamma)\) is a compact orientable 3-manifold \(M\) together with a set \(\gamma\) of disjoint annuli and tori in \(\partial M\). If there are no tori and the complement of the annuli in \(\partial M\) has no 2-sphere components then the notions of Heegaard splitting and Heegaard genus may be defined for such sutured manifolds in terms of decompositions into compression bodies. A Seifert surface for a link in \(S^ 3\) determines a complementary sutured manifold. The main results of this paper are concerned with the behaviour of the Heegaard genus \(h(M,\gamma)\) for such complementary sutured manifolds, with respect to Murasugi sum of Seifert surfaces. Examples are given to show that the result obtained is best possible. As a corollary it is shown that the Murasugi sum of fibre surfaces is again a fibre surface, a result first found by Stallings.