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Constructing Thompson representatives via pointed links

Susanna Terron

TL;DR

The work extends Jones' Thompson-link construction from $F$ to the Brown–Thompson group $F_3$, establishing a surjective map to pointed links via $\,\mathcal{L}_*$ and introducing the central monoid $(F_3,\diamond)$. It shows how connected sum and disjoint union operations on links correspond to the monoid and related actions on $F_3$, yielding a framework for standard forms of knot and tree-link representatives. The results provide a pathway toward a Markov-type theorem for Thompson representatives by analyzing prime decompositions through $\diamond$ and extend to wider families of links using disjoint union and linking moves. Overall, the paper builds a robust algebraic-topological bridge between $F_3$ and low-dimensional topology, enabling explicit Thompson representations for broad classes of links and tree links.

Abstract

We extend Jones' construction to obtain a surjective map from the Brown-Thompson group $F_3$ to the set of pointed links up to pointed isotopy. We then introduce an operation on $F_3$, and use it to define a new monoid $(F_3, \diamond)$, called the central monoid. Using the extended version of Jones' construction, we obtain a surjective monoid homomorphism from the central monoid to the monoid of pointed links with connected sum. This allows us to introduce a standard form for connected sum representatives in $F_3$, and we extend this construction to a certain family of links by defining disjoint union and linking moves on $F_3$.

Constructing Thompson representatives via pointed links

TL;DR

The work extends Jones' Thompson-link construction from to the Brown–Thompson group , establishing a surjective map to pointed links via and introducing the central monoid . It shows how connected sum and disjoint union operations on links correspond to the monoid and related actions on , yielding a framework for standard forms of knot and tree-link representatives. The results provide a pathway toward a Markov-type theorem for Thompson representatives by analyzing prime decompositions through and extend to wider families of links using disjoint union and linking moves. Overall, the paper builds a robust algebraic-topological bridge between and low-dimensional topology, enabling explicit Thompson representations for broad classes of links and tree links.

Abstract

We extend Jones' construction to obtain a surjective map from the Brown-Thompson group to the set of pointed links up to pointed isotopy. We then introduce an operation on , and use it to define a new monoid , called the central monoid. Using the extended version of Jones' construction, we obtain a surjective monoid homomorphism from the central monoid to the monoid of pointed links with connected sum. This allows us to introduce a standard form for connected sum representatives in , and we extend this construction to a certain family of links by defining disjoint union and linking moves on .

Paper Structure

This paper contains 14 sections, 24 theorems, 47 equations, 49 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Thompson's group
  3. Thompson's group F
  4. Brown--Thompson groups
  5. Jones' algorithm
  6. Extension to pointed link
  7. The central monoid
  8. Comparing operations
  9. Other monoids
  10. Connected sum and prime decomposition
  11. Prime decomposition
  12. Constructing link representatives
  13. Disjoint union
  14. Linking components

Key Result

Theorem A

The map $\mathcal{L}_*\colon F_3 \to \mathsf{Links}_*$ is surjective.

Figures (49)

  • Figure 1: The element $x_0\in F$, its dyadic subdivision, and the corresponding pair of trees.
  • Figure 2: Equivalence relation on binary trees. The left hand pair of trees is known as a caret.
  • Figure 3: The generators $y_0, y_1$ and $y_2$ of $F_3$ and the corresponding pairs of ternary trees.
  • Figure 4: The image of $x_0$ in $F_3$.
  • Figure 5: The link obtained from $x_0\in F$ via Jones' algorithm. Notice this is the unknot.
  • ...and 44 more figures

Theorems & Definitions (57)

  • Theorem A
  • Theorem B
  • Theorem C
  • Example 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 3.7
  • Lemma 3.8
  • proof
  • Lemma 3.9
  • ...and 47 more