Algebraic & Geometric Topology

Twisted quandle homology theory and cocycle knot invariants

J Scott Carter, Mohamed Elhamdadi, and Masahico Saito

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The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group cocycles. Explicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The corresponding generalization of the quandle cocycle knot invariants is given, by using the Alexander numbering of regions in the definition of state-sums. The invariants are used to derive information on twisted cohomology groups.

Article information

Algebr. Geom. Topol., Volume 2, Number 1 (2002), 95-135.

Received: 27 September 2001
Accepted: 8 February 2002
First available in Project Euclid: 21 December 2017

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Zentralblatt MATH identifier

Primary: 57N27 57N99: None of the above, but in this section
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25} 57T99: None of the above, but in this section

quandle homology cohomology extensions dihedral quandles Alexander numberings cocycle knot invariants


Carter, J Scott; Elhamdadi, Mohamed; Saito, Masahico. Twisted quandle homology theory and cocycle knot invariants. Algebr. Geom. Topol. 2 (2002), no. 1, 95--135. doi:10.2140/agt.2002.2.95.

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  • E Brieskorn, Automorphic sets and singularities, Contemporary Math., 78 (1988) 45–115.
  • K H Brown, Cohomology of groups, Graduate Texts in Mathematics, 87. Springer-Verlag, New York-Berlin (1982).
  • J S Carter, M Elhamdadi, M A Nikiforou, M Saito, Extensions of quandles and cocycle knot invariants, preprint.
  • J S Carter, D Jelsovsky, S Kamada, L Langford, M Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, to appear Trans AMS, preprint.
  • J S Carter, D Jelsovsky, S Kamada, M Saito, Computations of quandle cocycle invariants of knotted curves and surfaces, Advances in Math., 157 (2001) 36–94.
  • J S Carter, D Jelsovsky, S Kamada, M Saito, Quandle homology groups, their betti numbers, and virtual knots, J. of Pure and Applied Algebra, 157, (2001), 135–155.
  • J S Carter, S Kamada, M Saito, Alexander numbering of knotted surface diagrams, Proc. A.M.S. 128 no 12 (2000) 3761–3771.
  • J S Carter, S Kamada, M Saito, Geometric interpretations of quandle homology, J. of Knot Theory and its Ramifications, 10, no. 3 (2001) 345–386.
  • J S Carter, S Kamada, M Saito, Diagrammatic Computations for Quandles and Cocycle Knot Invariants, preprint.
  • J S Carter, M Saito, Canceling branch points on the projections of surfaces in 4-space, Proc. A. M. S. 116, 1, (1992) 229–237.
  • J S Carter, M Saito, Knotted surfaces and their diagrams, the American Mathematical Society, (1998).
  • V G Drinfeld, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) 798–820, Amer. Math. Soc., Providence, RI, 1987.
  • R Fenn, C Rourke, Racks and links in codimension two, Journal of Knot Theory and Its Ramifications Vol. 1 No. 4 (1992) 343–406.
  • R Fenn, C Rourke, B J Sanderson, Trunks and classifying spaces, Appl. Categ. Structures 3 no 4 (1995) 321–356.
  • R Fenn, C Rourke, B J Sanderson, James bundles and applications, preprint found at${}^{\sim}$cpr/ftp/
  • J Flower, Cyclic Bordism and Rack Spaces, Ph.D. Dissertation, Warwick (1995).
  • R H Fox, A quick trip through knot theory, in Topology of $3$-Manifolds, Ed. M.K. Fort Jr., Prentice-Hall (1962) 120–167.
  • M Gerstenhaber, S D Schack, Bialgebra cohomology, deformations, and quantum groups. Proc. Nat. Acad. Sci. U.S.A. 87 no. 1 (1990) 478–481.
  • M T Greene, Some Results in Geometric Topology and Geometry, Ph.D. Dissertation, Warwick (1997).
  • D Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Alg., 23, 37–65.
  • L H Kauffman, Knots and Physics, World Scientific, Series on knots and everything, vol. 1, 1991.
  • M Markl, J D Stasheff, Deformation theory via deviations. J. Algebra 170 no. 1 (1994) 122–155.
  • S Matveev, Distributive groupoids in knot theory, (Russian) Mat. Sb. (N.S.) 119(161) no. 1 (1982) 78–88, 160.
  • D Rolfsen, Knots and Links, Publish or Perish Press, (Berkeley 1976).
  • C Rourke, B J Sanderson, There are two $2$-twist-spun trefoils, preprint.
  • V Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988) 527–553.
  • E C Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965) 471–495.