Lorenz Links that are not Horseshoe and Rossler Links

Abstract

We compare the periodic orbit types of the Lorenz, horseshoe, and R\"ossler systems through their associated templates. Lorenz links are carried by the Lorenz template, while horseshoe links are carried by the horseshoe template. For the standard template model considered here, the R\"ossler template is identified, up to inversion symmetry, with the template of the horseshoe mechanism. Thus, in this paper, R\"ossler links and horseshoe links are treated as belonging to the same template class. We show that the relationship between Lorenz links and horseshoe/R\"ossler links has two complementary sides. First, we prove that the overlap between the two families is nontrivial by constructing infinite families of links which can be embedded in both templates. This verifies, for these families, a conjecture stated by Kofman that horseshoe links should also be Lorenz links. On the other hand, we prove that the two families are far from being the same. Holmes and Williams showed that many Lorenz torus knots cannot be embedded in the horseshoe template: if the torus knot \(T(p,q)\), with \(p<q\), is a horseshoe knot, then \(3p\leq 2q\). We show that this phenomenon is much broader. We extend the Holmes--Williams obstruction from torus knots to torus links, and we construct infinitely many hyperbolic Lorenz knots and links, as well as infinitely many satellite Lorenz knots and links, which cannot be embedded in the horseshoe template. Consequently, these examples are Lorenz links which are not horseshoe links and hence not R\"ossler links.