Journal of the Mathematical Society of Japan

Sheet number and quandle-colored 2-knot


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A diagram of a 2-knot consists of a finite number of compact, connected surfaces called sheets. We prove that if a 2-knot admits a non-trivial coloring by some quandle, then any diagram of the 2-knot needs at least four sheets. Moreover, if a 2-knot admits a non-trivial 5- or 7-coloring, then any diagram needs at least five or six sheets, respectively.

Article information

J. Math. Soc. Japan Volume 61, Number 2 (2009), 579-606.

First available in Project Euclid: 13 May 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25}
Secondary: 57Q35: Embeddings and immersions

2-knot quandle sheet number diagram triple point


SATOH, Shin. Sheet number and quandle-colored 2-knot. J. Math. Soc. Japan 61 (2009), no. 2, 579--606. doi:10.2969/jmsj/06120579.

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  • H. Aiso, On classification of simply knotted spheres with at most five crossing circles (Japanese), Master's thesis, Tokyo University.
  • E. Artin, Zur Isotopie zweidimensionaler Fächen im $\mathbf{R}^{4}$, Abh. Math. Sem. Univ. Hamburg, 4 (1925), 174–177.
  • J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc., 355 (2003), 3947–3989.
  • J. S. Carter, S. Kamada and M. Saito, Geometric interpretations of quandle homology and cocycle knot invariants, J. Knot Theory Ramifications, 10 (2001), 345–358.
  • J. S. Carter and M. Saito, Knotted surfaces and their diagrams, Mathematical Surveys and Monographs, 55, American Mathematical Society, Providence, RI, 1998.
  • D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Alg., 23 (1982), 37–65.
  • S. Matveev, Distributive groupoids in knot theory (Russian), Math. USSR-Sbornik, 46 (1982), 73–83.
  • M. Saito and S. Satoh, The spun trefoil needs four broken sheets, J. Knot Theory Ramifications, 14 (2005), 853–858.
  • S. Satoh, Sheet numbers of 2- and 3-twist-spun trefoils, preprint.
  • S. Satoh and A. Shima, The 2-twist-spun trefoil has the triple point number four, Trans. Amer. Math. Soc., 356 (2004), 1007–1024.
  • S. Satoh and A. Shima, Triple point numbers and quandle cocycle invariants of knotted surfaces in 4-space, New Zealand J. Math., 34 (2005), 71–79.