  A few years ago Vaughan Jones devised a method to construct knots and links
from elements of the Thompson groups. Given an element $g$ of $F_3$ we
construct a permutation $\mathcal{P}(g)$ which we call its Thompson
permutation. By analogy with the braid group, we show that the number of the
connected components of the link $\mathcal{L}(g)$ is equal to the number of
orbits of its Thompson permutation.
