Volume 7, issue 1 (2007)

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High-codimensional knots spun about manifolds

Dennis Roseman and Masamichi Takase

Algebraic & Geometric Topology 7 (2007) 359–377

arXiv: math/0609055


Using spinning we analyze in a geometric way Haefliger’s smoothly knotted (4k1)–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.

spinning, Haefliger knot, long knot, Seifert surface
Mathematical Subject Classification 2000
Primary: 57R40
Secondary: 57R65, 55P35
Received: 10 October 2006
Accepted: 20 November 2006
Published: 25 April 2007
Dennis Roseman
Department of Mathematics
University of Iowa
14 MacLean Hall
Iowa City IA 52242-1419
Masamichi Takase
Department of Mathematical Sciences
Faculty of Science
Shinshu University
Nagano 390-8621