Temperely-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 \( e^{2}_{i} = 2e_{i} ,{\kern 1pt} {\kern 1pt} e_{i} e_{j} e_{i} = e_{i} \) if |ij|=1 and e i e j =e j e i if |ij|≥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.

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Correspondence to Louis Kauffman*.

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* 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.

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Kauffman*, L., Thomas†, R. Temperely-Lieb Algebras and the Four-Color Theorem. Combinatorica 23, 653–667 (2003). https://doi.org/10.1007/s00493-003-0039-7

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Mathematics Subject Classification (2000):

  • 05C15
  • 82B20