Determination of generalized horseshoe maps inducing all link types

Let ϕ be an orientation preserving automorphism of the 2-disk, and Φ an isotopy from the identity to ϕ. Then for each periodic orbit P of ϕ, Φ uniquely determines a link SΦP in R3 by suspension. We say that ϕ induces all link types if there exists such an isotopy Φ that any link is realized as SΦP for some P. In this paper, we completely determine when generalized horseshoe maps induce all link types.