  We continue our study of the integer-valued knot invariants $\nu^\sharp(K)$
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
$\nu^\sharp(K)$ 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 $\epsilon^\sharp(K)$ 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 $S^3$.
