## $3$-manifolds with planar presentations and the width of satellite knots

HTML articles powered by AMS MathViewer

- by Martin Scharlemann and Jennifer Schultens PDF
- Trans. Amer. Math. Soc.
**358**(2006), 3781-3805 Request permission

## 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.** *The width of a satellite knot is no less than the width of its pattern knot and so $w(K_{1} \# K_{2}) \geq max(w(K_{1}), w(K_{2}))$. *

## References

- Gerhard Burde and Heiner Zieschang,
*Knots*, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR**808776** - Ralph H. Fox,
*On the imbedding of polyhedra in $3$-space*, Ann. of Math. (2)**49**(1948), 462–470. MR**26326**, DOI 10.2307/1969291 - David Gabai,
*Foliations and the topology of $3$-manifolds. II*, J. Differential Geom.**26**(1987), no. 3, 461–478. MR**910017** - W. B. Raymond Lickorish,
*An introduction to knot theory*, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, New York, 1997. MR**1472978**, DOI 10.1007/978-1-4612-0691-0 - Kanji Morimoto,
*There are knots whose tunnel numbers go down under connected sum*, Proc. Amer. Math. Soc.**123**(1995), no. 11, 3527–3532. MR**1317043**, DOI 10.1090/S0002-9939-1995-1317043-4 - Kanji Morimoto and Jennifer Schultens,
*Tunnel numbers of small knots do not go down under connected sum*, Proc. Amer. Math. Soc.**128**(2000), no. 1, 269–278. MR**1641065**, DOI 10.1090/S0002-9939-99-05160-6 - Yo’av Rieck and Eric Sedgwick,
*Thin position for a connected sum of small knots*, Algebr. Geom. Topol.**2**(2002), 297–309. MR**1917054**, DOI 10.2140/agt.2002.2.297 - Dale Rolfsen,
*Knots and links*, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR**0515288** - Martin Scharlemann,
*Handlebody complements in the $3$-sphere: a remark on a theorem of Fox*, Proc. Amer. Math. Soc.**115**(1992), no. 4, 1115–1117. MR**1116272**, DOI 10.1090/S0002-9939-1992-1116272-X - Martin Scharlemann and Jennifer Schultens,
*Annuli in generalized Heegaard splittings and degeneration of tunnel number*, Math. Ann.**317**(2000), no. 4, 783–820. MR**1777119**, DOI 10.1007/PL00004423 - M. Scharlemann, A. Thompson,
*On the additivity of knot width*, ArXiv preprint math.GT/0403326. - Horst Schubert,
*Über eine numerische Knoteninvariante*, Math. Z.**61**(1954), 245–288 (German). MR**72483**, DOI 10.1007/BF01181346 - Jennifer Schultens,
*Additivity of bridge numbers of knots*, Math. Proc. Cambridge Philos. Soc.**135**(2003), no. 3, 539–544. MR**2018265**, DOI 10.1017/S0305004103006832 - A. Thompson, personal communication.

## Additional Information

**Martin Scharlemann**- Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
- MR Author ID: 155620
- Email: mgscharl@math.ucsb.edu
**Jennifer Schultens**- Affiliation: Department of Mathematics, University of California, Davis, California 95616
- Email: jcs@math.ucdavis.edu
- 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.
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2218999