Journal of the Mathematical Society of Japan

A functor-valued extension of knot quandles

Tetsuya ITO

Full-text: Open access

Abstract

For an oriented knot $K$, we construct a functor from the category of pointed quandles to the category of quandles in three different ways. This functor-valued invariant of a knot is an extension of the knot quandle. We also extend the quandle cocycle invariants of knots by using these quandle-valued invariants, and study their properties.

Article information

Source
J. Math. Soc. Japan, Volume 64, Number 4 (2012), 1147-1168.

Dates
First available in Project Euclid: 29 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1351516771

Digital Object Identifier
doi:10.2969/jmsj/06441147

Mathematical Reviews number (MathSciNet)
MR2998919

Zentralblatt MATH identifier
1283.57012

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds

Keywords
quandle knot quandle homology cocycle invariant quandle invariant functor

Citation

ITO, Tetsuya. A functor-valued extension of knot quandles. J. Math. Soc. Japan 64 (2012), no. 4, 1147--1168. doi:10.2969/jmsj/06441147. https://projecteuclid.org/euclid.jmsj/1351516771


Export citation

References

  • N. Andruskiewitsch and M. Graña, From racks to pointed Hopf algebras, Adv. Math., 178 (2003), 177–243.
  • J. S. Birman, Braids, Links, and Mapping Class Groups, Ann. of Math. Stud., 82, Princeton University Press, 1974.
  • J. Crisp and L. Paris, Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups, Pacific J. Math., 221 (2005), 1–27.
  • J. Carter, M. Elhamdadi, M. Graña and M. Saito, Cocycle knot invariants from quandle modules and generalized quandle homology, Osaka J. Math., 42 (2005), 499–541.
  • J. 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.
  • M. Eisermann, Homological characterization of the unknot, J. Pure. Appl. Algebra, 177 (2003), 131–157.
  • D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure. Appl. Algebra, 23 (1982), 37–65.
  • S. V. Matveev, Distributive groupoids in knot theory (Russian), Mat. Sb. (N.S.), 119 (1982), 78–88.
  • M. Wada, Group invariants of links, Topology, 31 (1992), 399–406.