  Starting from the work by Jones on representations of Thompson's group $F$,
subgroups of $F$ with interesting properties have been defined and studied. One
of these subgroups is called the $3$-colorable subgroup $\mathcal{F}$, which
consists of elements whose ``regions'' given by their tree diagrams are
$3$-colorable. On the other hand, in his work on representations, Jones also
gave a method to construct knots and links from elements of $F$. Therefore it
is a natural question to explore a relationship between elements in
$\mathcal{F}$ and $3$-colorable links in the sense of knot theory. In this
paper, we show that all elements in $\mathcal{F}$ give 3-colorable links.
