The $p$-colorable subgroup of Thompson's group
Yuya Kodama, Akihiro Takano
TL;DR
This work defines the $p$-colorable subgroup $\mathcal{F}_p$ of Thompson's group $F$ for odd $p>2$ by comparing the dyadic-leaf coordinates of corresponding tree diagrams modulo $p$. It proves a precise isomorphism $\mathcal{F}_p \cong F(2^q)$ where $q$ is the order of $2$ in $(\mathbb{Z}/p\mathbb{Z})^ imes$, linking colorability to Brown–Thompson groups via an explicit embedding $\phi_q$. Using Jones' knot construction, the authors show that any nontrivial element of $\mathcal{F}_p$ yields a $p$-colorable knot/link and provide concrete examples (e.g., the trefoil for $p=3$, $7_7$ for certain $p$). They further interpret $\mathcal{F}_p$ as a stabilizer subgroup in a $q$-fold leaf-quotient, establishing that $\mathcal{F}_p$ is the stabilizer of a $q$-modulo-$p$ coloring and characterizing its generators in terms of $F(2^q)$ via $\phi_q$. Overall, the paper extends the 3-colorable framework to general odd primes, offering a cohesive algebraic and combinatorial perspective on $p$-colorability within Thompson knot theory and its Brown–Thompson counterparts.
Abstract
Recently, Jones introduced a method of constructing knots and links from elements of Thompson's group $F$ by using its unitary representations. He also defined several subgroups of $F$ as the stabilizer subgroups and some researchers studied them algebraically. One of the subgroups is called the 3-colorable subgroup $\mathcal{F}$, and the authors proved that all knots and links obtained from non-trivial elements of $\mathcal{F}$ are 3-colorable. In this paper, for any odd integer $p$ greater than two, we define the $p$-colorable subgroup of $F$ whose non-trivial elements yield $p$-colorable knots and links and show it is isomorphic to the certain Brown--Thompson group.