Heegaard分解在纽结Dehn手术和卫星结隧道数中的应用

三维流形的最小Heegaard亏格问题是三维流形研究的一个重要问题.由Lickorish-Wallace定理，任意一个三维流形都是三维球面中的某个链环的Dehn手术.因此一个自然的问题：沿纽结做Dehn手术得到的三维流形的最小Heegaard亏格是否小于纽结补空间的最小Heegaard亏格.三维球面中的纽结分类是纽结理论研究中的一个中心问题.一个自然的问题是列举所有具有相同隧道数的纽结.由于任意一个纽结要么是环面结，或卫星结，或双曲结，因此我们可以在该三个范畴下讨论上述问题.由于环面结的隧道数至多为1，因此我们考虑卫星结以及双曲结的隧道数问题. .本项目将研究三维流形Heegaard分解最小亏格在Dehn手术下的变化；借助于Heegaard分解的技巧，考虑缆绳结的隧道数与其伴随结之间的关系以及卫星结与其伴随结之间的关系.

The minimal Heegaard genus problem of a 3-manifold is an important question in the 3-manifold research. By Lickorish-Wallace’s theorem, any 3-manifold can be obtained by doing a Dehn surgery on a link in the 3-sphere. So there is a natural question: Whether the minimal Heegaard genus of the 3-manifold obtained by doing a Dehn surgery on a knot is less than that of the knot complement. Classing knots in the 3-sphere is a center problem of the knot theory. A natural question is to list all knots of the same tunnel number. Since any knot is a torus knot, or satellite knot, or hyperbolic knot, we can discuss the above problem in these three categories. As the tunnel number of a torus knot is at most one, we consider the tunnel numbers of satellite and hyperbolic knots..In this project, we will study the change of the minimal Heegaard genus of a 3-manifold under Dehn surgery; By the technique of Heegaard splitting, we will consider the relation between tunnel numbers of a cable knot and its companion, and the relation between a satellite knot and its companion.