Papers › A computational study of the number of connected components of positive Thompson links
A computational study of the number of connected components of positive Thompson links
Valeriano Aiello, Stefano Iovieno
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Almost a decade ago Vaughan Jones introduced a method to produce knots from elements of the Thompson groups F, which was later extended to the Brown-Thompson group F₃. In this article we define a way to produce permutations out of elements of the F and F₃ that we call Thompson permutations. The number of orbits of each Thompson permutation coincides with the number of connected components of the link. We explore the positive elements of F₃ of fixed \emph{width} and \emph{height} and make some conjectures based on numerical experiments. In order to define the Thompson permutations we need to assign an orientation to each link produced from elements of F and F₃. We prove that all oriented links can be produced in this way.
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