Idempotents in the Temperley-Lieb Monoid and Other Categories

Abstract: This paper examines idempotents in algebras and categories that arise from factorizations of the identity morphism. In the diagrammatic and combinatorial contexts considered here, these factorizations correspond to generalizations of meanders, where a meander is understood to be a curve in the plane that wanders transversely back and forth across a given straight line. By formulating the algebra of the Temperley-Lieb Monoid in terms of planar curve combinatorics, one can understand idempotents in the Temperley-Lieb Monoid in terms of meanders. Corresponding results are shown for the Brauer Monoid and for the Tangle Monoid and Tangle Category.