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A computational study of the number of connected components of positive Thompson links

Dec 2022

Almost a decade ago Vaughan Jones introduced a method to produce knots fromelements of the Thompson groups $F$, which was later extended to theBrown-Thompson group $F_3$. In this article we define a way to producepermutations out of elements of the $F$ and $F_3$ that we call Thompsonpermutations. The number of orbits of each Thompson permutation coincides withthe number of connected components of the link. We explore the positiveelements of $F_3$ of fixed \emph{width} and \emph{height} and make someconjectures based on numerical experiments. In order to define the Thompsonpermutations we need to assign an orientation to each link produced fromelements of $F$ and $F_3$. We prove that all oriented links can be produced inthis way.

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