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

TL;DR

The paper characterizes certain maximal subgroups of a Thompson group subgroup as stabilizers and analyzes how the -index of knots can increase by at most 3 when changing knot orientation.

Contribution

It identifies three maximal subgroups of infinite index as stabilizers and bounds the -index increase under orientation change.

Findings

Three maximal subgroups are stabilizers of certain actions.

The -index increases by at most 3 when changing knot orientation.

The -index is an elementary knot invariant related to Thompson groups.

Abstract

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.