  For a given smooth $2$-knot in $S^4$, we relate the existence of a smooth
Seifert hypersurface of a certain class to the existence of irreducible $
SU(2)$-representations of its knot group. For example, we see that any smooth
$2$-knot having the Poincar\'e homology $3$-sphere as a Seifert hypersurface
has at least four irreducible $SU(2)$-representations of its knot group. This
result is false in the topological category. The proof uses a quantitative
formulation of instanton Floer homology. Using similar techniques, we also
obtain similar results about codimension-$1$ embeddings of homology $3$-spheres
into closed definite $4$-manifolds and a fixed point type theorem for instanton
Floer homology.
