A Proof of the Pentagon Relation for Skeins

In \cite{HSZ23}, with Gus Schrader and Eric Zaslow we developed a skein-theoretic version of cluster theory, and made a conjecture on the pentagon relation for the skein dilogarithm. Here we give a topological proof of this conjecture. Combining \cite{MS21} and \cite{BCMN23}, we get a surjection from the skein algebra ${\mathrm{S}\mathrm{k}}^{+}(T-D)$ to the positive part of the elliptic Hall algebra ${\mathcal{E}}_{q,t}^{+}$. Hence our pentagon relation generalizes the ones in \cite{Z23} and \cite{GM19}.