Tokyo Journal of Mathematics

On Classification of Quandles of Cyclic Type

Seiichi KAMADA, Hiroshi TAMARU, and Koshiro WADA

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we study quandles of cyclic type, which form a particular subclass of finite quandles. The main result of this paper describes the set of isomorphism classes of quandles of cyclic type in terms of certain cyclic permutations. By using our description, we give a direct classification of quandles of cyclic type with cardinality up to 12.

Article information

Tokyo J. Math., Volume 39, Number 1 (2016), 157-171.

First available in Project Euclid: 30 March 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 53C35: Symmetric spaces [See also 32M15, 57T15]


KAMADA, Seiichi; TAMARU, Hiroshi; WADA, Koshiro. On Classification of Quandles of Cyclic Type. Tokyo J. Math. 39 (2016), no. 1, 157--171. doi:10.3836/tjm/1459367262.

Export citation


  • J. S. Carter, A Survey of Quandle Ideas, Introductory Lectures on Knot Theory, eds. L. H. Kauffman et al., Ser. Knots Everything 46, World Sci. Publ., 2012, 22–53.
  • 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.
  • R. Fenn and C. Rourke, Racks and links in codimension two, J. Knot Theory Ramifications 1 (1992), 343–406.
  • C. Hayashi, Canonical forms for operation tables of finite connected quandles, Comm. Algebra 41 (2013), 3340–3349.
  • A. Ishii, M. Iwakiri, Y. Jang and K. Oshiro, A $G$-family of quandles and handlebody-knots, Illinois J. Math. 57 (2013), 817–838.
  • D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), 37–65.
  • S. Kamada, Kyokumen musubime riron (Surface-knot theory) (in Japanese), Springer Gendai Sugaku Series 16, Maruzen Publishing Co. Ltd., 2012.
  • S. Kamada and K. Oshiro, Homology groups of symmetric quandles and cocycle invariants of links and surface-links, Trans. Amer. Math. Soc. 362 (2010), 5501–5527.
  • P. Lopes and D. Roseman, On finite racks and quandles, Comm. Algebra 34 (2006), 371–406.
  • T. Nosaka, Quandle cocycles from invariant theory, Adv. Math. 245 (2013), 423–438.
  • H. Tamaru, Two-point homogeneous quandles with prime cardinally, J. Math. Soc. Japan 65 (2013), 1117–1134.
  • L. Vendramin, On the classification of quandles of low order, J. Knot Theory Ramifications 21 (2012), 1250088, 10 pp.
  • L. Vendramin, Doubly transitive groups and cyclic quandles, J. Math. Soc. Japan, to appear.
  • K. Wada, Two-point homogeneous quandles with cardinality of prime power, Hiroshima Math. J. 45 (2015), 165–174.