A skein-valued lift of the unidirectional $A_n$-quiver $q$-dilogarithm relations

TL;DR

This paper extends $q$-dilogarithm identities related to unidirectional $A_n$-quivers into the HOMFLYPT-skein algebra of a genus $n$ handlebody, revealing new algebraic structures and relations.

Contribution

It introduces a skein-valued lift of $q$-dilogarithm identities, connecting quiver relations with skein algebras of handlebodies.

Findings

$q$-dilogarithm identities lift to skein algebra

Establishes connections between quiver relations and skein theory

Provides new algebraic tools for handlebody invariants

Abstract

We show that the $q$-dilogarithm identities associated to the unidirectional $A_n$-quiver lift to the HOMFLYPT-skein algebra of a genus $n$ handlebody.