Tunnel Numbers, Heegaard Genus and Generalized Primality

9803826 Schulten这个研究项目涉及Heegaard分裂与流形分解的关系，这种关系已经在特殊情况下以结点隧道数为幌子进行了研究。在连通节点和的运算下，节点的隧道数表现出相当不规律的行为。该项目需要对不可压缩表面的作用进行更全面的调查，因为它们与Heegaard分裂有关。特别地，它应该提供以下信息：(1)纽结隧道数相对于其广义素数分解的下界；(2)在连通纽结和下隧道数目退化的界；(3)流形的环面分解；(4)Haken流形的Heegaard亏格；以及(5)稳定化问题。这个项目的总体理念源于这样一种认识，即人们已经从几个角度研究了三维流形。其中两个观点涉及沿着曲面将3-歧管切割成基本的积木。沿着曲面切割以了解曲面所在的3-流形的想法，是沿着圆切割以了解圆所在的曲面的想法的自然延伸。在曲面的情况下，对曲面中包含的圆的分析就足以表征曲面。已经得到了一个完整的分类。3-流形已经被证明是远不那么容易处理的对象。事实上，将这一非常成功的想法推广到曲面分析中，在三维流形的研究中产生了两种截然不同的观点。调查者努力调和这两种观点。***

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. ***