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