Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T14:52:08.787Z Has data issue: false hasContentIssue false
  • English
  • Français

Unknotting sequences for torus knots

Published online by Cambridge University Press:  06 July 2009

SEBASTIAN BAADER
SEBASTIAN BAADER*
Affiliation:
Department of Mathematics, ETH Zürich, Switzerland. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The unknotting number of a knot is bounded from below by its slice genus. It is a well-known fact that the genera and unknotting numbers of torus knots coincide. In this paper we characterize quasipositive knots for which the genus bound is sharp: the slice genus of a quasipositive knot equals its unknotting number, if and only if the given knot appears in an unknotting sequence of a torus knot.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 1 , January 2010 , pp. 111 - 116
Copyright
Copyright © Cambridge Philosophical Society 2009

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]Baader, S.Slice and Gordian numbers of track knots. Osaka J. Math. 42 (2005), 257–271.Google Scholar
[2]Baader, S.Note on crossing changes. Q. J. Math. 57 (2006), 139–142.CrossRefGoogle Scholar
[3]Boileau, M. and Orevkov, S.Quasi-positivité d'une courbe analytique dans une boule pseudo-convexe. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 9, 825–830.CrossRefGoogle Scholar
[4]Boileau, M. and Weber, C.Le problème de J. Milnor sur le nombre gordien des nœuds algébriques. Enseign. Math. (2) 30 (1984), no. 3–4, 173–222.Google Scholar
[5]Hedden, M. and Ording, P. The Ozsváth-Szabó and Rasmussen concordance invariants are not equal. ArXiv: math.GT/0512348, 2005.Google Scholar
[6]Kawamura, T.On unknotting numbers and four-dimensional clasp numbers of links. Proc. Amer. Math. Soc. 130 (2002), no. 1, 243–252.CrossRefGoogle Scholar
[7]Kronheimer, P. B. and Mrowka, T. S.The genus of embedded surfaces in the projective plane. Math. Res. Lett. 1 (1994), no. 6, 797–808.CrossRefGoogle Scholar
[8]Nakamura, T.Four-genus and unknotting number of positive knots and links. Osaka J. Math. 37 (2000), no. 2, 441–451.Google Scholar
[9]Rasmussen, J. Khovanov homology and the slice genus. ArXiv: math.GT/0402131, 2004.Google Scholar
[10]Rolfsen, D.Knots and Links (Publish or Perish, 1976).Google Scholar
[11]Rudolph, L.Algebraic functions and closed braids. Topology 22 (1983), no. 2, 191–202.CrossRefGoogle Scholar
[12]Rudolph, L.Quasipositivity as an obstruction to sliceness. Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 51–59.CrossRefGoogle Scholar
[13]Rudolph, L.Positive links are strongly quasipositive. Proceedings of the Kirbyfest (Berkeley, CA, 1998), 555–562, Geom. Topol. Monogr. 2 (Geom. Topol. Publ., Coventry, 1999).CrossRefGoogle Scholar
[14]Wendt, H.Die gordische Auflösung von Knoten. Math. Z. 42 (1937), no. 1, 680–696.CrossRefGoogle Scholar