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We show that there is a unique Markov trace on the tower of Temperley--Liebtype quotients of Hecke algebras of Coxeter type $E_n$ (for all $n \geq 6$). Weexplain in detail how this trace may be computed easily using tom Dieck'scalculus of diagrams. As applications, we show how to use the trace to showthat the diagram representation is faithful, and to compute leadingcoefficients of certain Kazhdan--Lusztig polynomials.

On the Markov trace for Temperley--Lieb algebras of type $E_n$