纽结补中的不可压缩配对不可压缩曲面

Incompressible pairwise incompressible surfaces in knot complement

研究了曲面F与纽结补S^(3)-K中2维球面交拓扑图的性质,这些拓扑图是由一些环路和马鞍型圆盘组成.然后,给出了两个变换(即R-变换、S^(2)-变换)和连通和分解以及拓扑图的特征数,即E(T)=n_(+)+n_(-)-n_(s),而且这些变换不改变特征数.进而刻画环链补中不可压缩配对不可压缩曲面的性质,如果F∩S_(+)^(2)(or F∩S_(-)^(2))的分支数小于5并且交错纽结或几乎交错纽结的拓扑图是几乎简单时,曲面的亏格等于零.

In this paper,we discuss the properties that the surface F intersects with 2-spheres in S^(3)-K.The intersection forms a topological graph which consists of a collection of circles and saddle-shaped disks.Then we introduce the moves and define the characteristic number of the topological graph for F∩S_(+)^(2).The characteristic number is unchanged under the moves.By these ways,we characterize the properties of incompressible and pairwise incompressible surfaces in link exteriors.And then we prove that the genus of the surface equals zero if the complements number of F∩S_(+)^(2)(or F∩S_(-)^(2))is less than five and the graph is almost simple for alternating or almost alternating links.