On The Stabilization Problem For Heegaard Splittings Of Knot Exteriors

The stabilization of Heegaard splitting is one of the most important theories of Heegaard splitting. Generally, it may be difficult to find the reducing disks of a given Heegaard splitting. Fortunately, T. Kobayashi and T. Saito have given a necessary and sufficient condition for a stabilized Heegaard splitting of knot exteriors. In this article, we give a similar theorem based on the T. Kobayashi and T. Saito's.M. Scharlemann and J. Schultens proved that the tunnel number of the connected sum of n non-trivial knots is greater or equal to n. As an application, we give an example to support their idea that the equation can hold.Generally speaking, the union of two unkotting tunnel systems of a knot may be not a unknotting tunnel system, even if they are isotopic. In this article, we consider two unkotting tunnel systems for tunnel number one knot consisting of one and n arcs, respectively, and give a condition such that the union of them is still a unkotting tunnel system.