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There is a procedure, suggested by Vaughan Jones, which associates a link to every element of Thompson's group F. Also every knot or link in $\mathbb{R}^3$ can be obtained in this way. A subclass of links is the class of algebraic links.

My questions are the following: what type of elements of Thompson's F is correspond to algebraic links? What is the structure of the group generated by them? Is it elementary amenable? What is the impact of amenability or non-amenability of Thompson's F on the knot-links side?

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    $\begingroup$ A stupid answer. Assuming the construction is "canonical", any two conjugate elements yield isomorphic links. Hence, those elements corresponding to algebraic links generate a normal subgroup. And $F$ has very few normal subgroups: $\{1\}$, and those subgroups containing $[F,F]$ (hence not elementary amenable). $\endgroup$
    – YCor
    Commented Apr 9, 2021 at 16:22

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