Temperley-Lieb algebras and the four-color theorem

The Temperley-Lieb algebra T-n with parameter 2 is the associative algebra over Q generated by 1, e(0), e(1),...,e(n), where the generators satisfy the relations e(i)(2) = 2e(i), e(i)e(j)e(i) = e(i) if \i - j\ = 1 and e(i)e(j) = e(j)e(i) if \i -j\ greater than or equal to 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.