Temperley-lieb Algebras and the Four-color Theorem

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 ei = 2ei, eiejei = ei 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. 14 March 1999, revised 24 May 2001 Published in Combinatorica 23 (2003), 653–667. ∗ Partially supported by NSF under Grant DMS-9802859 and by NSA under grant MDA904-97-1-0015. ∗∗ Partially supported by NSF under Grant DMS-9623031 and by NSA under Grant MDA904-98-1-0517.