Temperely-Lieb Algebras and the Four-Color Theorem

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The Temperley–Lieb algebra Tn with parameter 2 is the associative algebra over Q generated by 1,e0,e1, . . .,en, where the generators satisfy the relations $$e^{2}_{i} = 2e_{i} ,{\kern 1pt} {\kern 1pt} e_{i} e_{j} e_{i} = e_{i}$$ if |i−j|=1 and eiej=ejei if |i−j|≥2. We use the Four Color Theorem to give a necessary and sufficient condition for certain elements of Tn to be nonzero. It turns out that the characterization is, in fact, equivalent to the Four Color Theorem.