Advertisement

On the Classification of f-Quandles

  • Indu Rasika Churchill
  • Mohamed ElhamdadiEmail author
  • Nicolas Van Kempen
Conference paper
  • 85 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 317)

Abstract

We use the structural aspects of the f-quandle theory to classify, up to isomorphisms, all f-quandles of order n. The classification is based on an effective algorithm that generate and check all f-quandles for a given order. We also include a pseudocode of the algorithm.

Keywords

Quandle f-quandle Isomorphism Structural aspect Classification 

References

  1. 1.
    Carter, J.S., Crans, A.S., Elhamdadi, M., Saito, M.: Cohomology of categorical self-distributivity. J. Homotopy Relat. Struct. 3(1), 13–63 (2008)Google Scholar
  2. 2.
    Carter, J.S., Crans, A., Elhamdadi, M., Gra\(\tilde{n}\)a, M., Saito, M.: Cocycle knot invariants from quandle modules and generalized quandle homology. Osaka J. Math 42(3), 499–541 (2005)Google Scholar
  3. 3.
    Churchill, I.R.U., Elhamdadi, M., Green, M., Makhlouf, A.: f-racks, f-quandles, their extensions and cohomolog. J. Algebra Appl. 16(6) (2017)Google Scholar
  4. 4.
    Churchill, I.R.U., Elhamdadi, M., Green, M., Makhlouf, A.: Ternary and \(n\)-ary \(f\)-distributive structures. Open Math. 16, 32–45 (2018)Google Scholar
  5. 5.
    Churchill, I.R.U., Elhamdadi, M., Fernando, N.: The cocycle structure of the Alexander \(f\)-quandles on finite fields. J. Algebra Appl. 17(10) (2018)Google Scholar
  6. 6.
    Elhamdadi, M., Moutuou, E.M.: Foundations of topological racks and quandles. J. Knot Theory Ramif. 25(3) (2016)Google Scholar
  7. 7.
    Elhamdadi, M., Nelson, S.: Quandles—an introduction to the algebra of knots. In: American Mathematical Society, Providence, RI, vol. 74 (2015)Google Scholar
  8. 8.
    Elhamdadi, M., Macquarrie, J., Restrepo, R.: Automorphism groups of quandles. J. Algebra Appl. 11(1) (2012)Google Scholar
  9. 9.
    Fenn, R.: Rourke, Colin: Racks and links in codimension two. J. Knot Theory Ramif. 1(4), 343–406 (1992)CrossRefGoogle Scholar
  10. 10.
    Hartwig, J.T., Larsson, D., Silvestrov, S.D.: Deformations of Lie algebras using \(\sigma \)-derivations. J. Algebra 295(2), 314–361 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Joyce, D.: A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23(1), 37–65 (1982)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Makhlouf, A., Silvestrov, S.D.: Hom-algebra structures. J. Gen. Lie Theory Appl. 2(2), 51–64 (2008)Google Scholar
  13. 13.
    Matveev, S.V.: Distributive groupoids in knot theory. Mat. Sb. (N.S.) 119(161), 78–88, 160 (1982)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Indu Rasika Churchill
    • 1
  • Mohamed Elhamdadi
    • 2
    Email author
  • Nicolas Van Kempen
    • 1
  1. 1.Department of MathematicsState University of New York at OswegoOswegoUSA
  2. 2.Department of MathematicsUniversity of South FloridaTampaUSA

Personalised recommendations