An introduction to Thompson knot theory and to Jones subgroups

TL;DR

This work surveys Vaughan Jones's program linking Thompson groups to knot theory, detailing two complementary knot-construction procedures—unoriented and oriented—involving $F$ and its Brown–Thompson analogues $F_k$. It introduces oriented subgroups $\vec{F}$ and $\vec{F}_3$ (and the $3$-colorable subgroup $\mathcal{F}$) via colorings of associated Tait graphs, establishes isomorphisms and stabilizer characterizations, and proves Alexander-type results ensuring Thompson-group elements can realize broad classes of links. The positive Thompson links are fully characterized as closures of bipartite arborescent tangles, linking positivity to arborescent structure, while Ren’s maps connect oriented subgroups to Brown–Thompson groups, revealing maximal-infinite-index subgroups in $F$ and $F_3$. The paper also highlights open questions about finiteness properties, presentations, and whether the $\mathcal{F}$ elements generate all unoriented knots. Overall, it extends the braid–knot paradigm to Thompson groups, offering new algebraic pathways to knot invariants and link classifications with potential implications for unitary representations and subgroup structure in Thompson groups.

Abstract

We review a constructions of knots from elements of the Thompson groups due to Vaughan Jones, which comes in two flavours: oriented and unoriented.

Figures (14)

• Figure 1: The generators of $F=F_2$.
• Figure 2: The generators of $F_3$.
• Figure 3: The generators of $F_4$.
• Figure 4: The local rules for computing the action of $F$ on numbers expressed in binary expansion.
• Figure 5: The local rules for computing the action of $F_3$ on numbers expressed in ternary expansion.

Theorems & Definitions (15)

• Remark 2.1
• Remark 2.2
• Example 2.3: The $4_1$ knot
• Theorem 2.4
• Theorem 2.5
• Theorem 3.1
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 5.1