High-codimensional knots spun about manifolds
Using spinning we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere. Consider the 2-torus standardly embedded in the 3-sphere, which is further standardly embedded in the 6-sphere. At each point of the 2-torus we have the normal disk pair: a 4-dimensional disk and a 1-dimensional proper sub-disk. We consider an isotopy (deformation) of the normal 1-disk inside the normal 4-disk, by using a map from the 2-torus to the space of long knots in 4-space, first considered by Budney. We use this isotopy in a construction called spinning about a submanifold introduced by the first-named author. Our main observation is that the resultant spun knot provides a generator of the Haefliger knot group of knotted 3-spheres in the 6-sphere. Our argument uses an explicit construction of a Seifert surface for the spun knot and works also for higher-dimensional Haefliger knots.