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