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Thompson's group $F$, tangles, and link homology

Vyacheslav Krushkal, Louisa Liles, Yangxiao Luo

Abstract

We extend a construction of Jones to associate $(n, n)$-tangles with elements of Thompson's group $F$ and prove that it is asymptotically faithful as $n \to\infty$. Using this construction we show that the oriented Thompson group $\vec F$ admits a lax group action on a category of Khovanov's chain complexes.

Thompson's group $F$, tangles, and link homology

Abstract

We extend a construction of Jones to associate -tangles with elements of Thompson's group and prove that it is asymptotically faithful as . Using this construction we show that the oriented Thompson group admits a lax group action on a category of Khovanov's chain complexes.
Paper Structure (12 sections, 13 theorems, 18 equations, 21 figures)

This paper contains 12 sections, 13 theorems, 18 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Background on Thompson's group $F$ and strand diagrams
  3. From elements of Thompson's groups $F$ to tangles
  4. The statement and the proof of faithfulness
  5. Orientability
  6. Oriented strand diagrams and the oriented Thompson group
  7. From oriented strand diagrams to oriented tangles
  8. A lax group action on Khovanov's chain complexes
  9. Definition of a lax group action
  10. Khovanov Chain Complexes
  11. Construction of the lax group action
  12. Verifying the axioms

Key Result

Theorem 1.1

Suppose $g, h \in F$ are two distinct elements. Then for sufficiently large $n$, the tangles $T_n(g), T_n(h)$ are not isotopic.

Figures (21)

  • Figure 1: An element $g \in F$ given by the ordered pair $(I,J)$ and its associated pair of trees.
  • Figure 2: The same element $g$ as in Figure \ref{['fig: thompson group trees']}, but the pair of partitions $(I',J')$ is a refinement of $(I,J)$ and the pair of trees differs by a cancelling caret, shown in red.
  • Figure 3: an example of a $(5,3)$-strand diagram
  • Figure 4: A split (left) and a merge (right).
  • Figure 5: Cancellation moves of type I (left) and Type II (right). Type I moves are applied when a split contains a merge directly below it, and the two share a pair of edges. Type II moves are applied when a merge occurs directly above a split, and the two share one edge.
  • ...and 16 more figures

Theorems & Definitions (52)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Proposition 3.4
  • proof
  • ...and 42 more