Lorenz links, T-links, Minimal Braids, Positive braids with a full twist, and Geometric Types

Abstract

Lorenz links and T-links are equivalent families by the work of Birman--Kofman. Lorenz links arise as periodic orbits of the Lorenz system, whereas T-links are closures of certain positive braids. Birman, Williams, and Franks showed that every Lorenz link admits a positive braid with a full twist realizing the braid index. We study Lorenz links from the minimal-braid viewpoint. We compute these representatives from the T-link parameters and show that they are precisely a family of positive braids with a full twist, called V-link braids. This gives an explicit correspondence between V-links and T-links. We use this correspondence to study equivalences among T-link presentations. We obtain criteria for distinguishing T-links from their parameters, recover the Birman--Kofman equivalence from the V-link/T-link correspondence, and prove that, under mild assumptions, this equivalence gives two distinct presentations of the same link. We also generalize this phenomenon: under natural non-degeneracy conditions, one obtains four distinct T-link presentations of the same link. More generally, even with the braid index fixed, the number of distinct V-link and T-link presentations can be arbitrarily large. In contrast, we initiate the study of uniqueness of T-link presentations. We prove that there are infinite families of Lorenz links with a unique T-link presentation, after excluding the destabilization case. For the Lorenz flow, this gives families of periodic orbits that admit only one T-link parameter description in their isotopy class. Finally, the V-link form places Lorenz links in the setting of positive braids with a full twist, where techniques for detecting geometric type apply. We prove criteria for such braids to be satellite or hyperbolic and apply them to V-links and T-links. As a consequence, we obtain new families of satellite and hyperbolic T-links and generalize known results.