纽结补中曲面的分段不可压缩性(英文)

Pairwise Incompressibility of Surfaces in Knot Complements

讨论了纽结补中的不可压缩、分段不可压缩曲面的性质.设K是S3中素的几乎交错纽结,F是S3-K中的不可压缩、分段不可压缩曲面,那么在F∩S2±中一定存在S2型和PS3型环路.通过研究F∩S2±中的环路性质,证明了对于固定的边界分之数,曲面类是有限(在同痕意义下),同时也证明了如果纽结K是两个排叉结的连通和,则曲面F是穿孔球面.

We study the properties of incompressible pairwise incompressible surfaces in knot exteriors. Let K be a prime almost alternating knot in S^3 and let F be an incompressible pairwise incompressible suface in S^3-K .Then there exist loops of type S^2 and of type PS^3 in F∩S^2_± .We prove that there are only finitely many such surfaces in S^3-K with n boundaries components for fixed n by discussing the properties of loops in F∩S^2_± ,and show that F is punctured sphere if K is a connected sum of two pretzel knots.