Kauffman bracket skein module

For an oriented 3-manifold, take the linear span of isotopy classes of framed links, including the empty link, and impose the local bracket smoothing relation and the trivial-circle relation with factor $-A^2-A^{-2}$. Fixing framed boundary endpoints gives its relative version. Gluing manifolds or stacking cylinders induces maps between skein modules; framed isotopy, rather than unframed Reidemeister-I invariance, is built into this construction.

The relative skein module fixes finitely many framed endpoints on the boundary of a 3-manifold and uses properly embedded framed tangles with those endpoints. Local Kauffman bracket relations are imposed away from the boundary. Gluing compatible endpoint sets composes tangles; a rectangle cylinder with n endpoints on each horizontal side gives the Temperley-Lieb diagram algebra.