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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
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
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Abstract

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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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Andruskiewitsch, N. and Graña, M., ‘From racks to pointed Hopf algebras’, Adv. Math. 178 (2003), 177243.CrossRefGoogle Scholar
[2]Asami, S. and Satoh, S., ‘An infinite family of non-invertible surfaces in 4-space’, Bull. London Math. Soc. 37 (2005), 285296.CrossRefGoogle Scholar
[3]Boyle, J., ‘Classifying 1-handles attached to knotted surfaces’, Trans. Amer. Math. Soc. 306 (1988), 475487.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), 39473989.CrossRefGoogle Scholar
[5]Cochran, T. D., ‘Ribbon knots in S4’, J. London Math. Soc. 28 (1983), 563576.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), 313324.CrossRefGoogle Scholar
[8]Gordon, C. McA., ‘Ribbon concordance of knots in the 3-sphere’, Math. Ann. 257 (1981), 157170.CrossRefGoogle Scholar
[9]Joyce, D., ‘A classifying invariant of knots, the knot quandle’, J. Pure Appl. Alg. 23 (1982), 3765.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. 208228.CrossRefGoogle Scholar
[11]Matveev, S., ‘Distributive groupoids in knot theory’, Math. USSR-Sbornik 47 (1982), 7383 in Russian.CrossRefGoogle Scholar
[12]Miyazaki, K., ‘Ribbon concordance does not imply a degree one map’, Proc. Amer. Math. Soc. 108 (1990), 10551058.CrossRefGoogle Scholar
[13]Miyazaki, K., ‘Band-sums are ribbon concordant to the connected sum’, Proc. Amer. Math. Soc. 126 (1998), 34013406.CrossRefGoogle Scholar
[14]Mochizuki, T., ‘Some calculations of cohomology groups of finite Alexander quandles’, J. Pure Appl. Algebra 179 (2003), 287330.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. 275295.Google Scholar
[16]Satoh, S., ‘Surface diagrams of twist-spun 2-knots’, J. Knot Theory Ramifications 11 (2002), 413430.CrossRefGoogle Scholar
[17]Silver, D. S., ‘On knot-like groups and ribbon concordance’, J. Pure Appl. Algebra 82 (1992), 99105.CrossRefGoogle Scholar
[18]Zeeman, E. C., ‘Twisting spun knots’, Trans. Amer. Math. Soc. 115 (1965), 471495.CrossRefGoogle Scholar