A Proof of the Pentagon Relation for Skeins
Abstract
In 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 MS21 and BCMN23, we get a surjection from the skein algebra Sk+(T - D) to the positive part of the elliptic Hall algebra Eq, t+. Hence our pentagon relation generalizes the ones in Z23 and GM19.
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