Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T17:41:49.050Z Has data issue: false hasContentIssue false
  • English
  • Français

Ribbon concordance of surface-knots via quandle cocycle invariants

Part of: PL-topology

Published online by Cambridge University Press:  09 April 2009

J. Scott Carter ,
Masahico Saito  and
Shin Satoh
J. Scott Carter
Affiliation:
Department of Mathematics, University of South Alabama, Mobile AL 36688, USA, e-mail: carter@mathstat.usouthal.edu
Masahico Saito
Affiliation:
Department of Mathematics, University of South Florida, Tampa FL 33620, USA, e-mail: saito@math.usf.edu
Shin Satoh
Affiliation:
Department of Mathematics, Chiba University, Yayoi-cho 1–33, Inage-ku, Chiba 263-8522, Japan, e-mail: satoh@math.s.chiba-u.ac.jp
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.

We give necessary conditions of a surface-knot to be ribbon concordant to another, by introducing a new variant of the cocycle invariant of surface-knots in addition to using the invariant already known. We demonstrate that twist-spins of some torus knots are not ribbon concordant to their orientation reversed images.

Keywords

surface-knotribbon concordancequandlecocycle invarianttriple point.

MSC classification

Secondary: 57Q45: Knots and links (in high dimensions) 57Q35: Embeddings and immersions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 1 , February 2006 , pp. 131 - 147
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Andruskiewitsch, N. and Graña, M., ‘From racks to pointed Hopf algebras’, Adv. Math. 178 (2003), 177–243.CrossRefGoogle Scholar
[2]Asami, S. and Satoh, S., ‘An infinite family of non-invertible surfaces in 4-space’, Bull. London Math. Soc. 37 (2005), 285–296.CrossRefGoogle Scholar
[3]Boyle, J., ‘Classifying 1-handles attached to knotted surfaces’, Trans. Amer. Math. Soc. 306 (1988), 475–487.CrossRefGoogle Scholar
[4]Carter, J. S., Jelsovsky, D., Kamada, S., Langford, L. and Saito, M., ‘Quandle cohomology and statesum invariants of knotted curves and surfaces’, Trans. Amer. Math. Soc. 355 (2003), 3947–3989.CrossRefGoogle Scholar
[5]Cochran, T. D., ‘Ribbon knots in S4’, J. London Math. Soc. 28 (1983), 563–576.CrossRefGoogle Scholar
[6]Fenn, R., Rourke, C. and Sanderson, B., ‘James bundles and applications’, preprint, available http://www.maths.warwick.ac.uk/bjs/.Google Scholar
[7]Gilmer, P. M., ‘Ribbon concordance and a partial order on S-equivalence classes’, Topology Appl. 18 (1984), 313–324.CrossRefGoogle Scholar
[8]Gordon, C. McA., ‘Ribbon concordance of knots in the 3-sphere’, Math. Ann. 257 (1981), 157–170.CrossRefGoogle Scholar
[9]Joyce, D., ‘A classifying invariant of knots, the knot quandle’, J. Pure Appl. Alg. 23 (1982), 37–65.CrossRefGoogle Scholar
[10]Kawauchi, A., ‘Torsion linking forms on surface-knots and exact 4-manifolds’, in: Knots in Hellas'98 (Delphi), Ser. Knots Everything 24 (World Sci. Publishing, River Edge, NJ, 2000) pp. 208–228.CrossRefGoogle Scholar
[11]Matveev, S., ‘Distributive groupoids in knot theory’, Math. USSR-Sbornik 47 (1982), 73–83 in Russian.CrossRefGoogle Scholar
[12]Miyazaki, K., ‘Ribbon concordance does not imply a degree one map’, Proc. Amer. Math. Soc. 108 (1990), 1055–1058.CrossRefGoogle Scholar
[13]Miyazaki, K., ‘Band-sums are ribbon concordant to the connected sum’, Proc. Amer. Math. Soc. 126 (1998), 3401–3406.CrossRefGoogle Scholar
[14]Mochizuki, T., ‘Some calculations of cohomology groups of finite Alexander quandles’, J. Pure Appl. Algebra 179 (2003), 287–330.CrossRefGoogle Scholar
[15]Roseman, D., ‘Reidemeister-type moves for surfaces in four-dimensional space’, in: Knot Theory (Warsaw, 1995), Banach Center Publ. 42 (Polish Acad. Sci., Warsaw, 1998) pp. 275–295.Google Scholar
[16]Satoh, S., ‘Surface diagrams of twist-spun 2-knots’, J. Knot Theory Ramifications 11 (2002), 413–430.CrossRefGoogle Scholar
[17]Silver, D. S., ‘On knot-like groups and ribbon concordance’, J. Pure Appl. Algebra 82 (1992), 99–105.CrossRefGoogle Scholar
[18]Zeeman, E. C., ‘Twisting spun knots’, Trans. Amer. Math. Soc. 115 (1965), 471–495.CrossRefGoogle Scholar