A convergent rewriting system for the Temperley-Lieb algebra is exhibited whose normal forms are the classical Jones normal forms, but the oriented version's basis claim is left as a conjecture.
Oriented Temperley--Lieb algebras and combinatorial Kazhdan--Lusztig theory
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abstract
We define oriented Temperley--Lieb algebras for classical Hermitian symmetric spaces. This allows us to explain the existence of closed combinatorial formulae for the Kazhdan--Lusztig polynomials for these spaces.
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Search for a basis of the Temperley-Lieb algebra, using rewriting systems
A convergent rewriting system for the Temperley-Lieb algebra is exhibited whose normal forms are the classical Jones normal forms, but the oriented version's basis claim is left as a conjecture.