Remarks on some maximal subgroups of $F$ and on the $\vec{F}$-index of knots

We demonstrate that three maximal subgroups of infinite index in the rectangular subgroup $K_{(2,2)}$ of the Thompson group $F$, each containing Jones’s $3$-colorable subgroup $\mathcal{F}$, can be characterized as stabilizer subgroups. Additionally, we show that the $\vec{F}$-index, an elementary knot invariant introduced thanks to Jones’s construction of knots from Thompson groups, may increase at most by $3$ after changing the orientation of a knot.