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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Clark, W. Edwin Saito, Masahico and Vendramin, Leandro 2016. Quandle coloring and cocycle invariants of composite knots and abelian extensions. Journal of Knot Theory and Its Ramifications, Vol. 25, Issue. 05, p. 1650024.


    Nosaka, Takefumi 2011. On homotopy groups of quandle spaces and the quandle homotopy invariant of links. Topology and its Applications, Vol. 158, Issue. 8, p. 996.


    Tanaka, Kokoro 2007. Inequivalent surface-knots with the same knot quandle. Topology and its Applications, Vol. 154, Issue. 15, p. 2757.


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  • Currently known as: Journal of the Australian Mathematical Society Title history
    Journal of the Australian Mathematical Society, Volume 80, Issue 1
  • February 2006, pp. 131-147

Ribbon concordance of surface-knots via quandle cocycle invariants

  • J. Scott Carter (a1), Masahico Saito (a2) and Shin Satoh (a3)
  • DOI: http://dx.doi.org/10.1017/S1446788700011423
  • Published online: 01 April 2009
Abstract
Abstract

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.

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[1]N. Andruskiewitsch and M. Graña , ‘From racks to pointed Hopf algebras’, Adv. Math. 178 (2003), 177243.

[3]J. Boyle , ‘Classifying 1-handles attached to knotted surfaces’, Trans. Amer. Math. Soc. 306 (1988), 475487.

[4]J. S. Carter , D. Jelsovsky , S. Kamada , L. Langford and M. Saito , ‘Quandle cohomology and statesum invariants of knotted curves and surfaces’, Trans. Amer. Math. Soc. 355 (2003), 39473989.

[7]P. M. Gilmer , ‘Ribbon concordance and a partial order on S-equivalence classes’, Topology Appl. 18 (1984), 313324.

[8]C. McA. Gordon , ‘Ribbon concordance of knots in the 3-sphere’, Math. Ann. 257 (1981), 157170.

[9]D. Joyce , ‘A classifying invariant of knots, the knot quandle’, J. Pure Appl. Alg. 23 (1982), 3765.

[10]A. Kawauchi , ‘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.

[11]S. Matveev , ‘Distributive groupoids in knot theory’, Math. USSR-Sbornik 47 (1982), 7383 in Russian.

[12]K. Miyazaki , ‘Ribbon concordance does not imply a degree one map’, Proc. Amer. Math. Soc. 108 (1990), 10551058.

[13]K. Miyazaki , ‘Band-sums are ribbon concordant to the connected sum’, Proc. Amer. Math. Soc. 126 (1998), 34013406.

[14]T. Mochizuki , ‘Some calculations of cohomology groups of finite Alexander quandles’, J. Pure Appl. Algebra 179 (2003), 287330.

[16]S. Satoh , ‘Surface diagrams of twist-spun 2-knots’, J. Knot Theory Ramifications 11 (2002), 413430.

[17]D. S. Silver , ‘On knot-like groups and ribbon concordance’, J. Pure Appl. Algebra 82 (1992), 99105.

[18]E. C. Zeeman , ‘Twisting spun knots’, Trans. Amer. Math. Soc. 115 (1965), 471495.

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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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