Knot surgery formulae for instanton Floer homology II: applications

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 $I^{\mathrm{♯}}(S_0^3(K))$. 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 $\dim I^{\mathrm{♯}}(S_n^3(K))=n+2$ 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.