Remark 4.8. Similarly to the 2-handle formula from Theorem 3.2, the 1-handle formula from Theorem 4.7 expresses the skein module of the more complicated manifold as a quotient of a (countable) direct sum of invariants of simpler manifolds. A possibly relevant difference, however, is that the 2-handle formula features only finitely many summands with a given shift in quantum grading, whereas this number is infinite for the 1-handle formula.
The skein modules that have been computed using only the 2-handle formula, first and foremost in [25], are locally finite-dimensional, i.e. finite-dimensional in each bidegree. It is an open question whether this is true for all four-manifolds admitting handle decompositions without 1-handles. In the rest of this paper we will see that local finite-dimensionality may fail when 1-handles are present.