Framed instanton homology and concordance, II

We continue our study of the integer-valued knot invariants ${\nu }^{\mathrm{♯}}(K)$ and $r_0(K)$, which together determine the dimensions of the framed instanton homologies of all nonzero Dehn surgeries on $K$. We first establish a “conjugation” symmetry for the decomposition of cobordism maps constructed in our earlier work, and use this to prove, among many other things, that ${\nu }^{\mathrm{♯}}(K)$ is always either zero or odd. We then apply these technical results to study linear independence in the homology cobordism group, to define an instanton Floer analogue ${\epsilon }^{\mathrm{♯}}(K)$ of Hom’s $\epsilon$-invariant in Heegaard Floer homology, and to the problem of characterizing a given 3-manifold as Dehn surgery on a knot in $S^3$.