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A few years ago Vaughan Jones devised a method to construct knots and linksfrom elements of the Thompson groups. Given an element $g$ of $F_3$ weconstruct a permutation $\mathcal{P}(g)$ which we call its Thompsonpermutation. By analogy with the braid group, we show that the number of theconnected components of the link $\mathcal{L}(g)$ is equal to the number oforbits of its Thompson permutation.

The number of connected components of links associated with the Thompson group