Tunnel Numbers, Heegaard Genus and Generalized Primality

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9803826 Schultens This research project concerns the relation of Heegaard splittings to manifold decompositions, a relation that has been studied in a particular case under the guise of tunnel numbers of knots. The tunnel number of a knot has been shown to behave quite erratically under the operation of connected sum of knots. The project entails a more general investigation into the role of incompressible surfaces as they relate to Heegaard splittings. In particular, it should provide information on (1) lower bounds of the tunnel number of knots relative to their generalized prime decompositions; (2) bounds on the degeneration of tunnel number under connected sum of knots; (3) torus decompositions of manifolds; (4) the Heegaard genus of Haken manifolds; and (5) stabilization problems. The general philosophy of this project stems from a realization that 3-dimensional manifolds have been studied from several viewpoints. Two of these viewpoints involve cutting the 3-manifold along surfaces into basic building blocks. The idea of cutting along a surface to understand the 3-manifold in which the surface lies, is a natural extension of the idea of cutting along circles to understand the surface in which the circles lie. In the case of surfaces, the analysis of the circles contained in the surface suffices to characterize the surface. A complete classification has been obtained. 3-manifolds have proven to be far less tractable objects. In fact, extending the idea that has been so successful for the analysis of surfaces has engendered two very different viewpoints in the study of 3-manifolds. The investigator endeavors to reconcile these two viewpoints. ***