Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T19:38:34.548Z Has data issue: false hasContentIssue false
  • English
  • Français

Computable bounds for Rasmussen’s concordance invariant

Part of: Low-dimensional topology

Published online by Cambridge University Press:  13 December 2010

Andrew Lobb
Andrew Lobb*
Affiliation:
Mathematics Department, Stony Brook University, Stony Brook, NY 11794, USA (email: lobb@math.sunysb.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given a diagram D of a knot K, we give easily computable bounds for Rasmussen’s concordance invariant s(K). The bounds are not independent of the diagram D chosen, but we show that for diagrams satisfying a given condition the bounds are tight. As a corollary we improve on previously known Bennequin-type bounds on the slice genus.

Keywords

knotsslice genusslice-Bennequin

MSC classification

Secondary: 57M25: Knots and links in $S^3$
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 2 , March 2011 , pp. 661 - 668
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Beliakova, A. and Wehrli, S., Categorification of the colored Jones polynomial and the Rasmussen invariant of links, Canad. J. Math., to appear.Google Scholar
[2]Kawamura, T., The Rasmussen invariants and the sharper slice-Bennequin inequality on knots, Topology 46 (2007), 29–38.CrossRefGoogle Scholar
[3]Lee, E. S., An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), 554–586.CrossRefGoogle Scholar
[4]Plamenevskaya, O., Transverse knots and Khovanov homology, Math. Res. Lett. (2006), 571–586.CrossRefGoogle Scholar
[5]Rasmussen, J., Khovanov homology and the slice genus, Invent. Math. 182 (2010), 419–447.CrossRefGoogle Scholar
[6]Shumakovitch, A., Rasmussen invariant, slice-Bennequin inequality, and sliceness of knots, J. Knot Theory Ramifications (2007), 1403–1412.CrossRefGoogle Scholar
[7]Stallings, J., Constructions of fibered knots and links, in Algebraic and geometric topology, Proceedings of Symposia in Pure Mathematics, vol. XXXII.2 (American Mathematical Society, Providence, RI, 1978), 55–60.CrossRefGoogle Scholar
[8]Stoimenow, A., Some examples related to knot sliceness, J. Pure Appl. Algebra 210 (2007), 161–175.CrossRefGoogle Scholar
[9]Traczyk, P., A combinatorial formula for the signature of alternating diagrams, Fund. Math. 184 (2004), 311–316.CrossRefGoogle Scholar