We find a way to calculate the term with lowest exponent in the Kauffman bracket polynomial of a closed 3-string braid. This leads us to a new proof of the faithfulness of the Temperley-Lieb representation of the 3-string braid group B 3 . In particular we will prove that for any link with braid index 3 either the coefficient of the lowest degree term or the coefficient of the highest degree term of the Jones polynomial is equal to ±1.
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