No Access

PULL BACK RELATION FOR NON-SPHERICAL KNOTS

Département de Mathématiques, Université Louis Pasteur Strasbourg I, 7, rue René Descartes, 67084 Strasbourg cedex, France

Search for more papers by this author

YUKIO MATSUMOTO

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

Search for more papers by this author

, and

OSAMU SAEKI

Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan

Search for more papers by this author

https://doi.org/10.1142/S0218216504003378Cited by:0 (Source: Crossref)

Abstract

We introduce a new relation for high dimensional non-spherical knots, which is motivated by the codimension two surgery theory: a knot is a pull back of another knot if the former is obtained as the inverse image of the latter by a certain degree one map between the ambient spheres. We show that this relation defines a partial order for (2n-1)-dimensional simple fibered knots for n≥3. We also give some related results concerning cobordisms and isotopies of knots together with several important explicit examples.

Keywords:

AMSC: Primary 57Q45, Secondary 57R40, Secondary 57R65

References

V. Blanlœil and F. Michel, Comment. Math. Helv. 72, 30 (1997). Crossref, Web of Science, Google Scholar
H. Duan and S. Wang, Math. Z. 244, 67 (2003). Crossref, Web of Science, Google Scholar
A. Durfee, Topology 13, 47 (1974). Crossref, Google Scholar
M. Kato, J. Math. Soc. Japan 26, 454 (1974). Crossref, Web of Science, Google Scholar
L. Kauffman, Topology 13, 143 (1974). Crossref, Google Scholar
J. Levine, Comment. Math. Helv. 44, 229 (1969). Crossref, Web of Science, Google Scholar
S. López de Medrano, Actes du Congrès International des Mathématiciens (Gauthier-Villars, Paris, 1971) pp. 99–112. Google Scholar
Y. Matsumoto, Note on the splitting problem in codimension two, unpublished, circa 1973 . Google Scholar
J. Milnor and D. Husemoller , Symmetric Bilinear Forms ( Springer-Verlag , New York, Heidelberg , 1973 ) . Crossref, Google Scholar
R. Vogt, Knot Theory, Lecture Notes in Mathematics 685 (Springer-Verlag, Berlin, 1978) pp. 218–226. Crossref, Google Scholar
R. Vogt , Cobordismus von hochzusammenhängenden Knoten, Dissertation 116 ( Universität Bonn, Mathematisches Institut , Bonn , 1980 ) . Google Scholar