This is a companion paper to earlier work of the authors, which proved an integral surgery formula for framed instanton homology. First, we present an enhancement of the large surgery formula, a rational surgery formula for null-homologous knots in any 3-manifold, and a formula encoding a large portion of \(I^\sharp (S^3_0(K))\) . Second, we use the integral surgery formula to study the framed instanton homology of many 3-manifolds: Seifert fibered spaces with nonzero orbifold degrees, especially nontrivial circle bundles over any orientable surface, surgeries on a family of alternating knots and all twisted Whitehead doubles, and splicings with twist knots. Finally, we use the previous techniques and computations to study almost L-space knots, i.e., the knots \(K\subset S^3\) with \(\dim I^\sharp (S_n^3(K))=n+2\) for some \(n\in \mathbb {N}_+\) . We show that an almost L-space knot of genus at least 2 is fibered and strongly quasi-positive, and a genus-one almost L-space knot must be either the figure eight or the mirror of the \(5_2\) knot in Rolfsen’s knot table.
