Isotopy of Links

Abstract

For a link L in an open, orientable, triangulable three-manifold M, let F(M-L, M) denote the fundamental group of the complement, and let K_q(M-L,M) denote the q-th lower central subgroup of the kernel of the inclusion homomorphism F(M-L) → F(M). Then it is proved that the group F(M-L)/K_q(M-L,M) is invariant under isotope of L, where q denotes an arbitrary positive integer. (This generalizes a theorem of K. T. Chen.) Let α₁∈ F(M-L)/K_q(M-L,M) be a meridian to the i-th component of L, and let β₁ be a parallel. Then it is also proved that the conjugate classes of the α₁ and β₁ are isotopy invariants.