Using spinning we analyze in a geometric way Haefliger’s smoothly knotted
–spheres in the
–sphere. Consider
the
–torus standardly
embedded in the
–sphere,
which is further standardly embedded in the
–sphere. At each point of the
–torus we have the normal
disk pair: a
–dimensional
disk and a
–dimensional
proper sub-disk. We consider an isotopy (deformation) of the normal
–disk inside the normal
–disk, by using a
map from the
–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
–spheres in
the
–sphere.
Our argument uses an explicit construction of a Seifert surface for the spun knot and
works also for higher-dimensional Haefliger knots.
Keywords
spinning, Haefliger knot, long knot, Seifert surface