The $p$-colorable subgroup of Thompson’s group

Abstract

V. F. R. 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 a certain Brown–Thompson group.