  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 $\vec{F}$ 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 $\vec{F}$, answering a
question asked by Jones in 2018.
