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On Thompson knot theory and conjugacy classes of Thompson's group $F$

Yuanyuan Bao, Xiaobing Sheng

TL;DR

This work investigates how conjugacy classes in Thompson's group $F$ interact with Jones's construction that associates unoriented links to elements of $F$. Using annular strand diagrams and the Belk–Matucci framework, it proves that every unoriented link $L$ can be obtained from elements of $F$ drawn from distinct conjugacy classes, yielding a sequence of distinct conjugacy classes all producing $L$ (Theorem 1). It also demonstrates that within the conjugacy classes of the generators $x_0$ and $x_1$, one can construct infinitely many distinct links, notably a family of $2$-bridge links, by explicit tree-diagram manipulations and their associated Tait graphs. The results illuminate a Markov-type phenomenon for $F$-generated links and pave the way for further questions about selecting conjugacy classes to realize arbitrary links and subfamilies of links arising from fixed conjugacy classes.

Abstract

Jones introduced a method to produce unoriented links from elements of the Thompson's group $F$, and proved that any link can be produced by this construction. In this paper, we attempt to investigate the relations between conjugacy classes of the group $F$ and the links being constructed. For each unoriented link $L$, we find a sequence of elements of $F$ from distinct conjugacy classes which yield $L$ via Jones's construction. We also show that a sequence of $2$-bridge links can be constructed from elements in the conjugacy class of $x_0$ (resp. $x_1$).

On Thompson knot theory and conjugacy classes of Thompson's group $F$

TL;DR

This work investigates how conjugacy classes in Thompson's group interact with Jones's construction that associates unoriented links to elements of . Using annular strand diagrams and the Belk–Matucci framework, it proves that every unoriented link can be obtained from elements of drawn from distinct conjugacy classes, yielding a sequence of distinct conjugacy classes all producing (Theorem 1). It also demonstrates that within the conjugacy classes of the generators and , one can construct infinitely many distinct links, notably a family of -bridge links, by explicit tree-diagram manipulations and their associated Tait graphs. The results illuminate a Markov-type phenomenon for -generated links and pave the way for further questions about selecting conjugacy classes to realize arbitrary links and subfamilies of links arising from fixed conjugacy classes.

Abstract

Jones introduced a method to produce unoriented links from elements of the Thompson's group , and proved that any link can be produced by this construction. In this paper, we attempt to investigate the relations between conjugacy classes of the group and the links being constructed. For each unoriented link , we find a sequence of elements of from distinct conjugacy classes which yield via Jones's construction. We also show that a sequence of -bridge links can be constructed from elements in the conjugacy class of (resp. ).

Paper Structure

This paper contains 11 sections, 10 theorems, 1 equation, 23 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Thompson's group $F$
  4. Jones's construction
  5. Constructing Thompson Tait graphs
  6. An equivalent way of the construction
  7. Conjugacy classes of $F$ and annular strand diagrams
  8. A sequence of elements from distinct conjugacy classes that produce the same link.
  9. Distinct links constructed from the same conjugacy class of $F$
  10. The proof for the case of $x_0$
  11. The proof for the case of $x_1$.

Key Result

Theorem 2.1

The set of equivalent classes of tree diagrams together with the multiplication described is isomorphic to the Thompson's group $F.$

Figures (23)

  • Figure 1: A tree pair aligned vertically to form a tree diagram.
  • Figure 2: An elementary expansion.
  • Figure 3: Replacing the local diagram on the left by two vertical lines.
  • Figure 4: The multiplication of two tree diagrams.
  • Figure 5: The infinite generating set of $F$. The number of leaves in $x_i$ is $i+3$.
  • ...and 18 more figures

Theorems & Definitions (18)

  • Theorem 2.1: Thompson's group $F$
  • Theorem 2.2: Jones key3986040m
  • Example 2.3
  • Definition 2.4: Strand diagram Belk2014
  • Theorem 2.5: Belk2014
  • Definition 2.6: Annular strand diagram Belk2014
  • Theorem 2.7: Belk2014
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 8 more