2006/09/30 by Dennis Roseman, Masamichi Takase · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Isotopy #Knot (papermaking) #Spinning #Submanifold #math.AT #math.GT #msc:55P35 #msc:57R40 #msc:57R65
paper · pdf · doi:10.2140/agt.2007.7.359
published in Algebraic & Geometric Topology 7, 359-377 (Mathematical Sciences Publishers) · 14 pages, 10 figures
arxiv created 2006/10/08 · openalex publication_date 2007/04/25 · arxiv updated 2014/09/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/08
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.