#### Volume 7, issue 1 (2007)

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### Dennis Roseman and Masamichi Takase

Algebraic & Geometric Topology 7 (2007) 359–377
 arXiv: math/0609055
##### Abstract

Using spinning we analyze in a geometric way Haefliger’s smoothly knotted $\left(4k-1\right)$–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.

##### Keywords
spinning, Haefliger knot, long knot, Seifert surface
##### Mathematical Subject Classification 2000
Primary: 57R40
Secondary: 57R65, 55P35