  Almost a decade ago Vaughan Jones introduced a method to produce knots from
elements of the Thompson groups $F$, which was later extended to the
Brown-Thompson group $F_3$. In this article we define a way to produce
permutations out of elements of the $F$ and $F_3$ that we call Thompson
permutations. The number of orbits of each Thompson permutation coincides with
the number of connected components of the link. We explore the positive
elements of $F_3$ of fixed \emph{width} and \emph{height} and make some
conjectures based on numerical experiments. In order to define the Thompson
permutations we need to assign an orientation to each link produced from
elements of $F$ and $F_3$. We prove that all oriented links can be produced in
this way.
