Algebraic & Geometric Topology 7 (2007) 359–377

DOI: 10.2140/agt.2007.7.359

arXiv: math/0609055

Abstract

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.

Keywords

spinning, Haefliger knot, long knot, Seifert surface

Mathematical Subject Classification

Primary: 57R40

Secondary: 55P35, 57R65

References

Publication

Received: 10 October 2006
Accepted: 20 November 2006
Published: 25 April 2007

Authors