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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$3$-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
Published electronically: May 26, 2005
MathSciNet review: 2218999
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider compact $3$-manifolds $M$ having a submersion $h$ to $R$ in which each generic point inverse is a planar surface. The standard height function on a submanifold of $S^{3}$ is a motivating example. To $(M, h)$ we associate a connectivity graph $\Gamma$. For $M \subset S^{3}$, $\Gamma$ is a tree if and only if there is a Fox reimbedding of $M$ which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of $S^{3} - M$ is a tree, then there is a level-preserving reimbedding of $M$ so that $S^{3} - M$ is a connected sum of handlebodies.

Corollary.

$\bullet$ The width of a satellite knot is no less than the width of its pattern knot and so

$\bullet$ $w(K_{1} \char93 K_{2}) \geq max(w(K_{1}), w(K_{2}))$.


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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: http://dx.doi.org/10.1090/S0002-9947-05-03767-0
PII: S 0002-9947(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.