## Abstract

Recently Vaughan Jones showed that the R. Thompson group F encodes in a natural way all knots and links in R^{3}, and a certain subgroup F→ of F encodes all oriented knots and links. We answer several questions of Jones about F→. In particular we prove that the subgroup F→ is generated by x_{0}x_{1}, x_{1}x_{2}, x_{2}x_{3} (where x_{i}, i∈N are the standard generators of F) and is isomorphic to F_{3}, the analog of F where all slopes are powers of 3 and break points are 3-adic rationals. We also show that F→ coincides with its commensurator. Hence the linearization of the permutational representation of F on F/F→ is irreducible. We show how to replace 3 in the above results by an arbitrary n, and to construct a series of irreducible representations of F defined in a similar way. Finally we analyze Jones' construction and deduce that the Thompson index of a link is linearly bounded in terms of the number of crossings in a link diagram.

Original language | English |
---|---|

Pages (from-to) | 122-159 |

Number of pages | 38 |

Journal | Journal of Algebra |

Volume | 470 |

DOIs | |

State | Published - 15 Jan 2017 |

Externally published | Yes |

## Keywords

- Diagram groups
- Knots and links
- R. Thompson group
- Tree-diagrams

## ASJC Scopus subject areas

- Algebra and Number Theory