Sheet number and quandle-colored 2-knot
Shin SATOH
Source: J. Math. Soc. Japan Volume 61, Number 2
(2009), 579-606.
Abstract
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.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1242220723
Digital Object Identifier: doi:10.2969/jmsj/06120579
Zentralblatt MATH identifier: 05573651
Mathematical Reviews number (MathSciNet): MR2532902
References
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.
Mathematical Reviews (MathSciNet): MR1990571
Zentralblatt MATH: 1028.57003
Digital Object Identifier: doi:10.1090/S0002-9947-03-03046-0
JSTOR: links.jstor.org
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.
Mathematical Reviews (MathSciNet): MR1825963
Zentralblatt MATH: 1002.57019
Digital Object Identifier: doi:10.1142/S0218216501000901
J. S. Carter and M. Saito, Knotted surfaces and their diagrams, Mathematical Surveys and Monographs, 55, American Mathematical Society, Providence, RI, 1998.
Mathematical Reviews (MathSciNet): MR1487374
Zentralblatt MATH: 0904.57010
D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Alg., 23 (1982), 37–65.
Mathematical Reviews (MathSciNet): MR638121
Zentralblatt MATH: 0474.57003
Digital Object Identifier: doi:10.1016/0022-4049(82)90077-9
S. Matveev, Distributive groupoids in knot theory (Russian), Math. USSR-Sbornik, 46 (1982), 73–83.
Mathematical Reviews (MathSciNet): MR672410
M. Saito and S. Satoh, The spun trefoil needs four broken sheets, J. Knot Theory Ramifications, 14 (2005), 853–858.
Mathematical Reviews (MathSciNet): MR2187601
Zentralblatt MATH: 1090.57017
Digital Object Identifier: doi:10.1142/S0218216505004123
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.
Mathematical Reviews (MathSciNet): MR1984465
Zentralblatt MATH: 1037.57018
Digital Object Identifier: doi:10.1090/S0002-9947-03-03181-7
JSTOR: links.jstor.org
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.
Mathematical Reviews (MathSciNet): MR2141479
Zentralblatt MATH: 1084.57022
Journal of the Mathematical Society of Japan
