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Annular Links from Thompson's Group $T$

Louisa Liles

TL;DR

The paper extends Jones' program of encoding links by Thompson's groups from the planar setting ($F$) to the annular setting ($T$). It develops two equivalent annular constructions $\mathcal{L}_{\mathbb{A}}(g)$ from $g\in T$ and introduces Annular Thompson Badness (ATB) to reduce arbitrary edge-signed annular graphs to realizable ones via $2$-equivalence, thereby proving that every edge-signed annular graph arises from some $g\in T$. It also shows that the Jones polynomial of annular links appears as a coefficient of unitary representations of the oriented subgroup $\vec{T}$ at specific root-of-unity values, mirroring Aiello–Conti's results for $F$ and $T$. The work forges a bridge between annular knot theory, representation theory, and the combinatorics of Thompson groups, with explicit demonstrations such as the positive trefoil example, and points toward connections with categorification and planar algebras.

Abstract

In 2014 Jones showed how to associate links in the $3$-sphere to elements of Thompson's group $F$. We provide an analogue of this program for annular links and Thompson's group $T$. The main result is that any edge-signed graph embedded in the annulus is the Tait graph of an annular link built from an element of $T$. In analogy to the work of Aiello and Conti, we also show that the coefficients of certain unitary representations of $T$ recover the Jones polynomial of annular links.

Annular Links from Thompson's Group $T$

TL;DR

The paper extends Jones' program of encoding links by Thompson's groups from the planar setting () to the annular setting (). It develops two equivalent annular constructions from and introduces Annular Thompson Badness (ATB) to reduce arbitrary edge-signed annular graphs to realizable ones via -equivalence, thereby proving that every edge-signed annular graph arises from some . It also shows that the Jones polynomial of annular links appears as a coefficient of unitary representations of the oriented subgroup at specific root-of-unity values, mirroring Aiello–Conti's results for and . The work forges a bridge between annular knot theory, representation theory, and the combinatorics of Thompson groups, with explicit demonstrations such as the positive trefoil example, and points toward connections with categorification and planar algebras.

Abstract

In 2014 Jones showed how to associate links in the -sphere to elements of Thompson's group . We provide an analogue of this program for annular links and Thompson's group . The main result is that any edge-signed graph embedded in the annulus is the Tait graph of an annular link built from an element of . In analogy to the work of Aiello and Conti, we also show that the coefficients of certain unitary representations of recover the Jones polynomial of annular links.
Paper Structure (12 sections, 7 theorems, 12 equations, 16 figures)

This paper contains 12 sections, 7 theorems, 12 equations, 16 figures.

Table of Contents

  1. Introduction and Statement of Main Results
  2. Thompson's Groups $F$ and $T$
  3. Thompson's Group $F$
  4. Link Diagrams in the plane from $F$
  5. Thompson Badness
  6. The oriented subgroup $\vec{F}$
  7. Thompson's Group $T$
  8. Building Annular Links from Thompson's Group $T$
  9. Annular Thompson Badness
  10. Proof of Theorem \ref{['thm: main']}
  11. $2$-equivalence
  12. Example: a positive trefoil embedded in $\mathbb{A}$

Key Result

Theorem 1.1

Let $\Gamma \hookrightarrow \mathbb{A}$ be an edge-signed embedded graph. Then there exists some $g \in T$ such that $\mathcal{L}_{\mathbb{A}}(g)$ is isotopic in $\mathbb{A} \times I$ to $L_{\mathbb{A}}(\Gamma)$.

Figures (16)

  • Figure 1: An annular link built from an edge-signed graph embedded in $\mathbb{A}$.
  • Figure 2: A pair of standard dyadic partitions, their corresponding trees $R$ and $S$, and their associated element $g \in F$.
  • Figure 3: A pair-of-trees representation of the same element $g$ from Figure \ref{['fig:reduced_g']}, which differs from the pair in Figure \ref{['fig:reduced_g']} by a cancelling caret.
  • Figure 4: Diagrammatic composition in $F$ as given by belkgubasapir.
  • Figure 5: A Hopf link created from an element of $F$ via the construction introduced by Jones jones14.
  • ...and 11 more figures

Theorems & Definitions (12)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 3.1
  • Proposition 3.2
  • Definition
  • Proposition 3.3
  • proof
  • Lemma 4.1
  • proof
  • Remark
  • ...and 2 more