2.2. Ribbon Higman–Thompson groups. For convenience, we will think of the forest $F_+$ drawn beneath $F_-$ and upside down, i.e., with the root at the bottom and the leaves at the top. The permutation $\rho$ is then indicated by arrows pointing from the leaves of $F_-$ to the corresponding paired leaves of $F_+$. See Figure 2 for this visualization of (the unreduced representation of) the element of $V_{3,2}$ from Figure 1.
FIGURE 2. An element of $V_{3,2}$.
Now in the ribbon version of the Higman–Thompson groups, the permutations of leaves are simply replaced by ribbon braids which can twist between the leaves.
Definition 2.2. Let $\mathcal{I} = \Pi_{i=1}^d I_i : [0,1] \times {1, \dots, l} \to \mathbb{R}^2$ be an embedding which we refer to as the marked bands. A ribbon braid is a map $R : ([0,1] \times {0,1,\dots,l}) \times [0,1] \to \mathbb{R}^2$ such that for any $0 \le t \le 1$, $R_t : [0,1] \times {1,\dots,l} \to \mathbb{R}^2$ is an embedding, $R_0 = \mathcal{I}$, and there exists $\sigma \in S_l$ such that $R_1(t)|{I_i} = I{\sigma(i)}(t)$ or $R_1(t)|{I_i} = I{\sigma(i)}(1-t)$. The usual product of paths defines a group structure on the set of ribbon braids up to homotopy among ribbon braids. This group, denoted by $RB_l$, does not depend on the choice of the marked bands and it is called the ribbon braid group with $l$ bands.
Remark 2.3. Note that $RB_l \cong \mathbb{Z}^l \ltimes B_l$ where the action of $B_l$ is induced by the symmetric group action on the coordinates of $\mathbb{Z}^l$.
Definition 2.4. A ribbon braided paired (d,r)-forest diagram is a triple $(F_-, r, F_+)$ consisting of two $(d,r)$-forests $F_-$ and $F_+$ both with $l$ leaves for some $l$ and a ribbon braid $r \in RB_l$ connecting the leaves of $F_-$ to the leaves of $F_+$.
The expansion and reduction rules for the ribbon braids just come from the natural way of splitting a ribbon band into $d$ components and the inverse operation to this. See Figure 3 for how to split a half twisted band when $d=2$. Note that not only are the two bands themselves twisted but the bands are also braided. Everything else will be the same as in the braided case, so we omit the details here. As usual, we define two ribbon braided paired forest diagrams to be equivalent if one is obtained from the other by a sequence of reductions or expansions. The multiplication operation $\ast$ on the equivalence classes is defined the same way as for $bV_{d,r}$. We direct the reader to [SW, Section 2]
FIGURE 3. Splitting a ribbon into 2 ribbons.
Definition 2.5. The ribbon Higman–Thompson group $RV_{d,r}$ (resp. the oriented ribbon Higman–Thompson group $RV_{d,r}^+$) is the group of equivalence classes of (resp. oriented) ribbon braided paired $(d,r)$-forests diagrams with the multiplication $\ast$.