Research on finite Dehn surgeries on knots and links

Dehn手术是一种利用3球中的结点和连接产生3流形的手术。已知任何闭合可定向的3流形都可以通过在3球中的连杆上进行Dehn手术得到。球面上的结可以分为三种类型；环面结，卫星结和双曲结。从某种意义上说，由于Dehn手术对环面结和卫星结的特点，现在最有趣的情况是双曲结的情况。通过W.Thurston对双曲结上的Dehn手术的研究，我们知道大多数双曲结上的Dehn手术产生双曲流形。对于双曲结的情况，已知双曲结上的例外Dehn手术的次数，即产生非双曲流形的双曲结上的Dehn手术的次数是有限的。因此，对这种特殊的手术进行定性是一个非常重要的问题。我们正在研究对蒙特西诺斯结的特殊手术。由于Montesinos节上的可约手术和环面手术已经被刻画，我们研究了Montesinos节上的有限手术和Seifert手术。特别地，我们研究了一类有限运算是否存在未定的Montesinos节，并得到了部分结果。在那里，我们使用了M.Culler和P.Shalen开发的一种方法，该方法使用了结补的基本群的特征变体。不幸的是，由于研究尚未结束，对此类手术的完整描述是下一个项目。我们还得到了蒙特西诺斯结外表面的基本面与其特征品种之间新的显著关系。

Dehn surgery is an operation yielding 3-manifolds using knots and links in the 3-sphere. It is known that any closed orientable 3-manifold can be obtained by a Dehn surgery on a link in the 3-sphere. Knots in the 3-sphere can be classified into three types ; torus knots, satellite knots and hyperbolic knots. Since Dehn surgeries on torus knots and satellite knots have been characterized in a sense, the most interesting case now is the hyperbolic knot case. By W.Thurston's research on Dehn surgeries on hyperbolic knots, we know that most Dehn surgeries on hyperbolic knots yield hyperbolic manifolds. For the hyperbolic knot case, the number of exceptional Dehn surgeries on a hyperbolic knot, i.e. Dehn surgeries on a hyperbolic knot which yield non-hyperbolic manifolds, is known to be finite. Hence characterizing such exceptional surgeries is a very important problem. We are studying such exceptional surgeries on Montesinos knots. Since reducible surgeries and toroidal surgeries on Montesinos knots have been characterized, we studied finite surgeries and Seifert surgeries on Montesinos knots. Especially we studied some classes of Montesinos knots for which the existence of finite surgeries is undetermined and we obtained partial results. There we used a method, developed by M.Culler and P.Shalen, which uses character varieties of fundamental groups of knot complements. Unfortunately as the study has not finished, the complete characterization of such surgeries is a next project. We also obtained a new significant relation between essential surfaces in Montesinos knot exteriors and their character varieties.