-manifolds with planar presentations and the width of satellite knots

Authors:
Martin Scharlemann and Jennifer Schultens

Journal:
Trans. Amer. Math. Soc. **358** (2006), 3781-3805

MSC (2000):
Primary 57M25, 57M27

DOI:
https://doi.org/10.1090/S0002-9947-05-03767-0

Published electronically:
May 26, 2005

MathSciNet review:
2218999

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider compact -manifolds having a submersion to in which each generic point inverse is a planar surface. The standard height function on a submanifold of is a motivating example. To we associate a connectivity graph . For , is a tree if and only if there is a Fox reimbedding of which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of is a tree, then there is a level-preserving reimbedding of so that is a connected sum of handlebodies.

**Corollary.**

* The width of a satellite knot is no less than the width of its pattern knot and so*

.

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Additional Information

**Martin Scharlemann**

Affiliation:
Department of Mathematics, University of California, Santa Barbara, California 93106

Email:
mgscharl@math.ucsb.edu

**Jennifer Schultens**

Affiliation:
Department of Mathematics, University of California, Davis, California 95616

Email:
jcs@math.ucdavis.edu

DOI:
https://doi.org/10.1090/S0002-9947-05-03767-0

Received by editor(s):
September 28, 2003

Received by editor(s) in revised form:
May 18, 2004

Published electronically:
May 26, 2005

Additional Notes:
The authors thank RIMS Kyoto, where this work was begun, Professor Tsuyoshi Kobayashi for inviting us to RIMS, Yo’av Rieck for helpful conversations there, and the NSF for partial support via grants DMS 0203680 and DMS 0104039. The second author also thanks the MPIM-Bonn for support.

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.