On the Alexander Theorem for the oriented Thompson group $\vec{F}$

Abstract

In [Jo14] and [Jo18] Vaughan Jones introduced a construction which yields oriented knots and links from elements of the oriented Thompson group $\vec{F}$. In this paper we prove, by analogy with Alexander's classical theorem establishing that every knot or link can be represented as a closed braid, that given an oriented knot/link $\vec{L}$, there exists an element $g$ in $\vec{F}$ whose closure $\vec{\mathcal{L}}(g)$ is $\vec{L}$.

arXiv commentTo appear in Algebraic & Geometric Topology. Corrected definition

References in corpus (4)

• A no-go theorem for the continuum limit of a periodic quantum spin chain
• The Homflypt polynomial and the oriented Thompson group
• The Jones polynomial and functions of positive type on the oriented Jones-Thompson groups $\vec{F}$ and $\vec{T}$
• Graph polynomials and link invariants as positive type functions on Thompson's group F

Cited by in corpus (10)

• On the $3$-colorable subgroup $\mathcal{F}$ and maximal subgroups of Thompson's group $F$
• Arborescence of positive Thompson links
• On the oriented Thompson subgroup ${\vec{F}}_3$ and its relatives in higher Brown-Thompson groups
• Positive oriented Thompson links
• An introduction to Thompson knot theory and to Jones subgroups
• The $3$-colorable subgroup of Thompson's group and tricolorability of links
• Remarks on some maximal subgroups of $F$ and on the $\vec{F}$-index of knots
• The planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part
• Virtual Thompson's group
• An extension of Krishnan's central limit theorem to the Brown-Thompson groups