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High-codimensional knots
spun about manifolds
Dennis Roseman and Masamichi Takase |
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Algebraic & Geometric Topology 7 (2007) 359–377
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arXiv: math/0609055 |
Abstract |
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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
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Mathematical Subject Classification 2000
Primary: 57R40
Secondary: 57R65, 55P35
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References |
Publication
Received: 10 October 2006
Accepted: 20 November 2006
Published: 25 April 2007
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Authors |
| Dennis Roseman | |
| Department of Mathematics University of Iowa 14 MacLean Hall Iowa City IA 52242-1419 USA |
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| http://www.math.uiowa.edu/~roseman/ | |
| Masamichi Takase | |
| Department of Mathematical
Sciences Faculty of Science Shinshu University Matsumoto Nagano 390-8621 Japan |
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