The Hecke algebra \(\mathcal H\) associated to a finite Weyl group \(W\) has basis \(T_w\) indexed by the elements of \(W\) and for all generators \(s\) of \(W\) we have \(T_sT_w=T_{sw}\), if \(l(sw)>l(w)\) and \(T_s^2=(q-1)T_s+qT_e\) (where \(e\) is the identity element of \(W\)). It is an algebra over \(A={\mathbb Q} (q^{1/2})\). An involution on \(A\) by \(\overline{q^{1/2}}=q^{-1/2}\) we extend to an involution \(i\) on \(\mathcal H\) by setting: \(i(\sum_w\alpha_wT_w)=\sum_w\bar \alpha_w(T_{w^{-1}})^{-1}\). The Kazhdan-Lusztig polynomials are determined uniquely by the following: For any \(w\in W\) there is a unique element \(C_w'\in {\mathcal H}\) such that \(C_w'=q^{-l(w)/2}\sum_{x\leq w}P_{x,w}T_x\) and \(i(C_w')=C_w'\), where \(P_{x,w}\) is a polynomial in \(q\) of degree at most \((l(w)-l(x)-1)/2\) for \(x<w\), \(P_{w,w}=1\) and \(P_{x,w}=0\), if \(x\not\leq w\). Theorem 1. Let \({\mathbf a}=s_{i_1}\dots s_{i_r}\) be a reduced expression for \(w\in S_n\). The following are equivalent: (1) \(w\) is 321-hexagon-avoiding. (2) \(P_{x,w}=\sum q^{d(\sigma)}\), where \(d(\sigma)\) is the defect statistic and the sum is over all masks \(\sigma\) on \(\mathbf a\) whose product is \(x\). (3) The Poincaré polynomial for the full intersection cohomology group of \(X_w\) is \(\sum_i \dim(IH^{2i}(X_w))q^i=(1+q)^{l(w)}\). (4) The Kazhdan-Lusztig basis element \(C_w'\) satisfies \(C_w'=C_{s_{i_1}}'\dots C_{s_{i_r}}'\). (5) The Bott-Samelson resolution of \(X_w\) is small. (6) \(IH_*(X_w)\simeq H_*(Y)\), where \(Y\) is the Bott-Samelson resolution of \(X_w\).