Volume 7, issue 1 (2007)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 21
Issue 4, 1595–2140
Issue 3, 1075–1593
Issue 2, 543–1074
Issue 1, 1–541

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
 
Other MSP Journals
High-codimensional knots spun about manifolds

Dennis Roseman and Masamichi Takase

Algebraic & Geometric Topology 7 (2007) 359–377

arXiv: math/0609055

Abstract

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.

Keywords
spinning, Haefliger knot, long knot, Seifert surface
Mathematical Subject Classification 2000
Primary: 57R40
Secondary: 57R65, 55P35
References
Publication
Received: 10 October 2006
Accepted: 20 November 2006
Published: 25 April 2007
Authors
Dennis Roseman
Department of Mathematics
University of Iowa
14 MacLean Hall
Iowa City IA 52242-1419
USA
http://www.math.uiowa.edu/~roseman/
Masamichi Takase
Department of Mathematical Sciences
Faculty of Science
Shinshu University
Matsumoto
Nagano 390-8621
Japan