  We show that Haefliger's differentiable (6,3)-knot bounds, in 6-space, a
4-manifold (a Seifert surface) of arbitrarily prescribed signature. This
implies, according to our previous paper, that the Seifert surface has been
prolonged in a prescribed direction near its boundary. This aspect enables us
to understand a resemblance between Ekholm-Szucs' formula for the Smale
invariant and Boechat-Haefliger's formula for Haefliger knots. As a
consequence, we show that an immersion of the 3-sphere in 5-space can be
regularly homotoped to the projection of an embedding in 6-space if and only if
its Smale invariant is even. We also correct a sign error in our previous
paper: "A geometric formula for Haefliger knots" [Topology 43 (2004)
1425-1447].

Geometric Topology;Algebraic Topology