Osaka Journal of Mathematics

On symplectic quandles

Esteban Adam Navas and Sam Nelson

Full-text: Open access

PDF File (262 KB)

Abstract

We study the structure of symplectic quandles, quandles which are also $R$-modules equipped with an antisymmetric bilinear form. We show that every finite dimensional symplectic quandle over a finite field $\mathbb{F}$ or arbitrary field $\mathbb{F}$ of characteristic other than 2 is a disjoint union of a trivial quandle and a connected quandle. We use the module structure of a symplectic quandle over a finite ring to refine and strengthen the quandle counting invariant.

Article information

Source
Osaka J. Math., Volume 45, Number 4 (2008), 973-985.

Dates
First available in Project Euclid: 26 November 2008

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1227708829

Mathematical Reviews number (MathSciNet)
MR2493966

Zentralblatt MATH identifier
1168.57011

Subjects
Primary: 176D99 57M27: Invariants of knots and 3-manifolds 55M25: Degree, winding number

Citation

Navas, Esteban Adam; Nelson, Sam. On symplectic quandles. Osaka J. Math. 45 (2008), no. 4, 973--985. https://projecteuclid.org/euclid.ojm/1227708829


Export citation

References

  • N. Andruskiewitsch and M. Graña: From racks to pointed Hopf algebras, Adv. Math. 178 (2003), 177--243.
    Mathematical Reviews (MathSciNet): MR1994219
    Digital Object Identifier: doi:10.1016/S0001-8708(02)00071-3
    Zentralblatt MATH: 1032.16028
  • W.A. Adkins and S.H. Weintraub: Algebra, an approach via module theory, Graduate Texts in Mathematics 136, Springer, New York, 1992.
    Mathematical Reviews (MathSciNet): MR1181420
  • E. Brieskorn: Automorphic sets and braids and singularities; in Braids (Santa Cruz, CA, 1986), Contemp. Math. 78, Amer. Math. Soc., Providence, RI, 1988, 45--115.
    Mathematical Reviews (MathSciNet): MR975077
  • 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
    Digital Object Identifier: doi:10.1090/S0002-9947-03-03046-0
    JSTOR: links.jstor.org
    Zentralblatt MATH: 1028.57003
  • G. Ehrman, A. Gurpinar, M. Thibault and D.N. Yetter: Toward a classification of finite quandles, arXiv:math.QA/0512135.
    Mathematical Reviews (MathSciNet): MR2414453
    Digital Object Identifier: doi:10.1142/S0218216508006270
    Zentralblatt MATH: 1148.57018
  • R. Fenn and C. Rourke: Racks and links in codimension two, J. Knot Theory Ramifications 1 (1992), 343--406.
    Mathematical Reviews (MathSciNet): MR1194995
    Digital Object Identifier: doi:10.1142/S0218216592000203
    Zentralblatt MATH: 0787.57003
  • D. Joyce: A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), 37--65.
    Mathematical Reviews (MathSciNet): MR638121
    Digital Object Identifier: doi:10.1016/0022-4049(82)90077-9
    Zentralblatt MATH: 0474.57003
  • L.H. Kauffman and V.O. Manturov: Virtual biquandles, Fund. Math. 188 (2005), 103--146.
    Mathematical Reviews (MathSciNet): MR2191942
    Digital Object Identifier: doi:10.4064/fm188-0-6
    Zentralblatt MATH: 1088.57006
  • S.V. Matveev: Distributive groupoids in knot theory, Math. USSR, Sb. 47 (1984), 73--83.
  • S. Nelson: Classification of finite Alexander quandles, Topology Proc. 27 (2003), 245--258.
    Mathematical Reviews (MathSciNet): MR2048935
  • S. Nelson: A polynomial invariant of finite quandles, J. Algebra Appl. 7 (2008), 263--273.
    Mathematical Reviews (MathSciNet): MR2417045
    Digital Object Identifier: doi:10.1142/S0219498808002801
    Zentralblatt MATH: 1152.57015
  • S. Nelson and C.-Y. Wong: On the orbit decomposition of finite quandles, J. Knot Theory Ramifications 15 (2006), 761--772.
    Mathematical Reviews (MathSciNet): MR2253834
    Digital Object Identifier: doi:10.1142/S0218216506004701
    Zentralblatt MATH: 1105.57006
  • M. Niebrzydowski and J.H. Przytycki: Burnside kei, Fund. Math. 190 (2006), 211--229.
    Mathematical Reviews (MathSciNet): MR2232860
    Digital Object Identifier: doi:10.4064/fm190-0-8
    Zentralblatt MATH: 1100.57011
  • M. Takasaki: Abstraction of symmetric transformation, Tohoku Math J. 49 (1943), 145--207, (in Japanese).
    Mathematical Reviews (MathSciNet): MR21002
  • D.N. Yetter: Quandles and monodromy, J. Knot Theory Ramifications 12 (2003), 523--541.
    Mathematical Reviews (MathSciNet): MR1985908
    Digital Object Identifier: doi:10.1142/S0218216503002597
    Zentralblatt MATH: 1051.57003

Osaka University and Osaka City University, Departments of Mathematics

Osaka Journal of Mathematics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more