A Disproof of Santharoubane's Conjecture on Presentations of Generic Skein Algebras

Abstract

Let $Σ$ be a compact connected oriented surface of genus at least $3$ with at most one boundary component. Santharoubane associated to certain presentations of the mapping class group modulo its center a finitely presented algebra equipped with a canonical surjection onto the generic Kauffman bracket skein algebra of $Σ$, and conjectured that a suitable choice yields an algebra isomorphic to the skein algebra. We show that every algebra arising from this construction admits an augmentation character, whereas the generic skein algebra of $Σ$ admits no unital character over $\mathbb Q(A)$. The latter obstruction follows from the intersection-one Dehn-twist identity together with a $4$-holed-sphere skein relation. Consequently, the conjectured isomorphism does not hold as stated.