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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: [email protected]
KAZUTO TAKAO
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka, 560-0043, Japan. e-mail: [email protected]
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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 

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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