(Disk, Essential surface) pairs of Heegaard splittings that intersect in one point
Abstract
We consider a Heegaard splitting M=H1 S H2 of a 3-manifold M having an essential disk D in H1 and an essential surface F in H2 with |D F|=1. (We require that boundary of F is in S when H2 is a compressionbody with non-empty "minus" boundary.) Let F be a genus g surface with n boundary components. From S, we obtain a genus g(S)+2g+n-2 Heegaard splitting M=H'1 S' H'2 by cutting H2 along F and attaching F × [0,1] to H1. As an application, by using a theorem due to Casson and Gordon, we give examples of 3-manifolds having two Heegaard splittings of distinct genera where one of the two Heegaard splittings is a strongly irreducible non-minimal genus splitting and it is obtained from the other by the above construction.
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