  A few years ago the so-called oriented subgroup $\vec F$ of the Thompson
group $F$ was introduced by V. Jones while investigating the connections
between subfactors and conformal field theories. In the coding of links and
knots by elements of $F$ it corresponds exactly to the oriented ones. Thanks to
the work of Golan and Sapir, $\vec F$ provided the first example of a maximal
subgroup of infinite index in $F$ different from the parabolic subgroups that
fix a point in $(0,1)$. In this paper we investigate possible analogues of
$\vec F$ in higher Thompson groups $F_k, k\geq 2$, with $F=F_2$, introduced by
Brown. Most notably, we study algebraic properties of the oriented subgroup
$\vec{F}_3$ of $F_3$, as described recently by Jones, and prove in particular
that it gives rise to a non-parabolic maximal subgroup of infinite index in
$F_3$ and that the corresponding quasi-regular representation is irreducible.
