Regular isotopy classes of link diagrams from Thompson's groups

In 2014, Vaughan Jones developed a method to produce links from elements of Thompson's group F, and showed that all links arise this way. He also introduced a subgroup F -> of F and a method to produce oriented links from elements of this subgroup. In 2018, Valeriano Aiello showed that all oriented links arise from this construction. We classify exactly those regular isotopy classes of links that arise from F, as well as exactly those regular isotopy classes of oriented links that arise from F ->, answering a question asked by Jones in 2018.