Knot theory of R-covered Anosov flows: homotopy versus isotopy of closed orbits

TL;DR

This paper investigates the knot properties of periodic orbits in R-covered Anosov flows on 3-manifolds, establishing homotopy implies isotopy and unknottedness of lifts, with additional results for atoroidal manifolds.

Contribution

It proves that homotopic periodic orbits are isotopic and their lifts are unknotted, providing new insights into the knot theory of Anosov flows.

Findings

Homotopic orbits are isotopic.

Lifts of orbits are unknotted.

Finer properties in atoroidal manifolds.

Abstract

In this article, we study the knots realized by periodic orbits of R-covered Anosov flows in compact 3-manifolds. We show that if two orbits are freely homotopic then in fact they are isotopic. We show that lifts of periodic orbits to the universal cover are unknotted. When the manifold is atoroidal, we deduce some finer properties regarding the existence of embedded cylinders connecting two given homotopic orbits.