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Let $(W,S)$ be an arbitrary Coxeter system. We introduce a family ofpolynomials, $\{ \tilde{\mathcal{R}}_{u,\underline{v}}(t)\}$, indexed by pairs$(u,\underline{v})$ formed by an element $u\in W$ and a (non-necessarilyreduced) word $\underline{v}$ in the alphabet $S$. The polynomial$\tilde{\mathcal{R}}_{u,\underline{v}}(t)$ is obtained by considering a certainsubset of Libedinsky's light leaves associated to the pair $(u,\underline{v})$.Given a reduced expression $\underline{v}$ of an element $v\in W$; we show that$\tilde{\mathcal{R}}_{u,\underline{v}}(t)$ coincides with the Kazhdan-- Lusztig$\tilde{R}$-polynomial $\tilde{R}_{u,v}(t)$. Using the diagrammatic approach,we obtain some closed formulas for $\tilde{R}$- polynomials.

Diagrammatics for Kazhdan-Lusztig R-polynomials