We develop some applications of certain algebraic and combinatorial
conditions on the elements of Coxeter groups, such as elementary proofs of the
positivity of certain structure constants for the associated Kazhdan–Lusztig
basis. We also explore some consequences of the existence of a Jones-type trace
on the Hecke algebra of a Coxeter group, such as simple procedures for
computing leading terms of certain Kazhdan–Lusztig polynomials.

The sequel to this paper is math.QA/0509363.