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This article concerns Fox colorings. We prove that if a link admits non-trivial $(2k+1)$-colorings, with prime $2k+1\ge 11$, it also admits non-trivial $(2k+1)$-colorings not involving colors $2k,2k-1$, nor $k$.

In a recent paper, Jones introduced a correspondence between elements of the Thompson group $F$ and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be reinterpreted as coefficients of certain unitary representations of $F$. We give a somewhat different and elementary proof of this fact for the Kauffman bracket evaluated at certain roots of unity by means of a statistical mechanics model interpretation. Moreover, by similar methods we show that, for some particular specializations of the variables, other familiar link invariants and graph polynomials, namely the number of $N$-colorings and the Tutte polynomial, can be viewed as positive definite functions on $F$.