对于任意给定的一个带边三维流形的Heegaard分解，一个自然的问题：什么时候，它的边界稳定化是非稳定化的？Zou-Guo-Qiu利用Heegaard距离给出了一个充分条件.通过利用三维流形的组合技巧，本项目希望给出一个判定Heegaard分解的边界稳定化是非稳定化的充要条件..分类三维球面中的纽结是纽结理论的一个中心问题.列举具有相同隧道数的纽结是分类纽结的一条途径。本项目主要研究卫星结中缆绳结的隧道数问题.由于研究Heegaard亏格是研究纽结隧道数的一个有效途径，本项目将应用Heegaard分解的理论去研究缆绳结与其伴随结隧道数之间的关系. 同时我们将结合Heegaard Floer同调的理论来处理一些问题.

For a Heegaard splitting of any bounded 3-manifold,a natural question: When the boundary stabilization of it is unstabilized? Zou-Guo-Qiu gave a sufficient condition by using Heegaard distance. Through combinatorial techniques of 3-manifolds, we want to give a sufficient and necessary condition for the boundary stabilization of a Heegaard splitting is unstabilized..Classifying knots in the 3-sphere is a center problem of the knot theory. Listing knots with the same tunnel numbers is a way to classify knots. We mainly study tunnel numbers of cable knots of satellite knots. Since studying Heegaard genus is an effective way of studying the tunnel number of a knot, we will apply the theory of Heegaard splittings to study the relation between tunnel numbers of a cable knot and its companion. Meanwhile we will combine the theory of Heegaard Floer homology to deal with some problems.