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The Kauffman bracket skein module    K  (  M  )    of a 3‐manifold   M   is the quotient of the    Q  (  A  )   ‐vector space spanned by isotopy classes of links in   M   by the Kauffman relations. A conjecture of Witten states that if   M   is closed, then    K  (  M  )    is finite dimensional. We introduce a version of this conjecture for manifolds with boundary and prove a stability property for generic Dehn filling of knots. As a result, we provide the first hyperbolic examples of the conjecture, proving that almost all Dehn fillings of any two‐bridge knot satisfy the conjecture. This paper relies extensively on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures.

Infinite families of hyperbolic 3‐manifolds with finite‐dimensional skein modules