Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-14T18:43:27.272Z Has data issue: false hasContentIssue false
  • English
  • Français

A locally minimal, but not globally minimal, bridge position of a knot

Published online by Cambridge University Press:  25 April 2013

MAKOTO OZAWA  and
KAZUTO TAKAO
MAKOTO OZAWA
Affiliation:
Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University, 1-23-1 Komazawa, Setagaya-ku, Tokyo, 154-8525, Japan. e-mail: w3c@komazawa-u.ac.jp
KAZUTO TAKAO
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka, 560-0043, Japan. e-mail: kazutotakao@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a locally minimal, but not globally minimal, bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 1 , July 2013 , pp. 181 - 190
Copyright
Copyright © Cambridge Philosophical Society 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Alexander, J. W. and Briggs, G. B.On types of knotted curves. Ann. of Math. (2) 28 (1926/27), no. 1–4, 562–586.CrossRefGoogle Scholar
[2]Birman, J. S.On the stable equivalence of plat representations of knots and links. Canad. J. Math. 28 (1976), no. 2, 264–290.CrossRefGoogle Scholar
[3]Bachman, D. and Schleimer, S.Distance and bridge position. Pacific J. Math. 219 (2005), no. 2, 221–235.CrossRefGoogle Scholar
[4]Coward, A. Algorithmically detecting the bridge number of hyperbolic knots. arXiv:0710.1262.Google Scholar
[5]Crowell, R. H. and Fox, R. H.Reprint of the 1963 original, Introduction to Knot Theory. Graduate Texts in Mathematics 57 (Springer-Verlag, New York-Heidelberg, 1977).CrossRefGoogle Scholar
[6]Gabai, D.Foliations and the topology of 3-manifolds. III. J. Differential Geom. 26 (1987), no. 3, 479–536.Google Scholar
[7]Hayashi, C.Stable equivalence of Heegaard splittings of 1-submanifolds in 3-manifolds. Kobe J. Math. 15 (1998), no. 2, 147–156.Google Scholar
[8]Hayashi, C. and Shimokawa, K.Heegaard splittings of the trivial knot. J. Knot Theory Ramifications 7 (1998), no. 8, 1073–1085.CrossRefGoogle Scholar
[9]Hayashi, C. and Shimokawa, K.Thin position of a pair (3-manifold, 1-submanifold). Pacific J. Math. 197 (2001), no. 2, 301–324.CrossRefGoogle Scholar
[10]Heath, D. J. and Kobayashi, T.Essential tangle decomposition from thin position of a link. Pacific J. Math. 179 (1997), no. 1, 101–117.CrossRefGoogle Scholar
[11]Johnson, J. and Tomova, M.Flipping bridge surfaces and bounds on the stable bridge number. Algebr. Geom. Topol. 11 (2011), no. 4, 1987–2005.CrossRefGoogle Scholar
[12]Otal, J.-P.Présentations en ponts du nœud trivial. C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 16, 553–556.Google Scholar
[13]Otal, J.-P.Presentations en Ponts des Nœuds Rationnels. Low-dimensional topology (Chelwood Gate, 1982), 143–160, London Math. Soc. Lecture Note Ser., 95 (Cambridge University Press, 1985).Google Scholar
[14]Ozawa, M.Bridge position and the representativity of spatial graphs. Topology Appl. 159 (2012), no. 4, 936–947.CrossRefGoogle Scholar
[15]Ozawa, M.Non-minimal bridge positions of torus knots are stabilized. Math. Proc. Camb. Phil. Soc. 151 (2011) 307–317.CrossRefGoogle Scholar
[16]Reidemeister, K.Elementare Begründung der Knotentheorie. Abh. Math. Sem. Univ. Hamburg 5 (1927) 24–32.CrossRefGoogle Scholar
[17]Scharlemann, M.Thin Position in the Theory of Classical Knots. Handbook of knot theory, 429–459 (Elsevier B. V., Amsterdam, 2005).CrossRefGoogle Scholar
[18]Scharlemann, M. and Tomova, M.Uniqueness of bridge surfaces for 2-bridge knots. Math. Proc. Camb. Phil. Soc. 144 (2008), no. 3, 639–650.CrossRefGoogle Scholar
[19]Schubert, H.Über eine numerische Knoteninvariante. Math. Z. 61 (1954), 245–288.CrossRefGoogle Scholar
[20]Schultens, J.Additivity of bridge numbers of knots. Math. Proc. Camb. Phil. Soc. 135 (2003), no. 3, 539–544.CrossRefGoogle Scholar
[21]Schultens, J.Width complexes for knots and 3-manifolds. Pacific J. Math. 239 (2009), no. 1, 135–156.CrossRefGoogle Scholar
[22]Takao, K.Bridge decompositions with distances at least two. Hiroshima Math. J. 42 (2012), no. 2, 161–168.CrossRefGoogle Scholar
[23]Tomova, M.Thin position for knots in a 3-manifold. J. Lond. Math. Soc. (2) 80 (2009), no. 1, 85–98.CrossRefGoogle Scholar
[24]Tomova, M.Multiple bridge surfaces restrict knot distance. Algebr. Geom. Topol. 7 (2007), 957–1006.CrossRefGoogle Scholar
[25]Zupan, A.Properties of knots preserved by cabling. Comm. Anal. Geom. 19 (2011), no. 3, 541–562.CrossRefGoogle Scholar
[26]Zupan, A.Unexpected local minima in the width complexes for knots. Algebr. Geom. Topol. 11 (2011), no. 2, 1097–1105.CrossRefGoogle Scholar