An effective proof of finiteness for Kauffman bracket skein modules

We prove a version of the finiteness conjecture for Kauffman bracket skein modules of $3$-manifolds with boundary, which was introduced by the second author in \cite{Det21}. In particular our methods, which are constructive, give an alternative proof of Witten’s finiteness conjecture for the Kauffman bracket skein modules of closed $3$-manifolds, which was originally proved in \cite{GJS19}. Moreover, as a corollary we show that the peripheral ideal of any link is non-empty, answering a question of Frohman, Gelca and Lofaro \cite{FGL02}.