High-codimensional knots spun about manifolds

Details High-codimensional knots spun about manifolds High-codimensional knots spun about manifolds

2006-09-03

arxiv.org/abs/math/0609055v2

Algebraic and Geometric Topology 7 (2007) pp.359-377

Mathematics

Geometric Topology

14 pages, 10 figures

Scientific paper

10.2140/agt.2007.7.359

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.

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