  In 2017, Jones studied the unitary representations of Thompson's group $F$
and defined a method to construct knots and links from $F$. One of his results
is that any knot or link can be obtained from an element of this group, which
is called Alexander's theorem. On the other hand, Thompson's group $F$ has many
subgroups and it is known that there exist various subgroups which satisfy or
do not satisfy Alexander's theorem. In this paper, we prove that almost all
stabilizer subgroups under the natural action on the unit interval satisfy
Alexander's theorem.
