High-codimensional knots spun about manifolds

Roseman, Dennis¹ ; Takase, Masamichi²

¹ Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City IA 52242-1419, USA
² Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto, Nagano 390-8621, Japan

Algebraic and Geometric Topology, Tome 7 (2007) no. 1, pp. 359-377

Résumé

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.

Détail

DOI : 10.2140/agt.2007.7.359

Keywords: spinning, Haefliger knot, long knot, Seifert surface