On quandle homology groups of Alexander quandles of prime order

Nosaka, Takefumi

doi:10.1090/S0002-9947-2013-05754-6

On quandle homology groups of Alexander quandles of prime order
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Trans. Amer. Math. Soc. 365 (2013), 3413-3436 Request permission

Abstract:

In this paper we determine the integral quandle homology groups of Alexander quandles of prime order. As a special case, this settles the delayed Fibonacci conjecture by M. Niebrzydowski and J. H. Przytycki from their 2009 paper. Further, we determine the cohomology group of the Alexander quandle and obtain relatively simple presentations of all higher degree cocycles which generate the cohomology group. Finally, we prove that the integral quandle homology of a finite connected Alexander quandle is annihilated by the order of the quandle.

References

Soichiro Asami and Shin Satoh, An infinite family of non-invertible surfaces in 4-space, Bull. London Math. Soc. 37 (2005), no. 2, 285–296. MR 2119028, DOI 10.1112/S0024609304003832
J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford, and Masahico Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3947–3989. MR 1990571, DOI 10.1090/S0002-9947-03-03046-0
J. Scott Carter, Seiichi Kamada, and Masahico Saito, Geometric interpretations of quandle homology, J. Knot Theory Ramifications 10 (2001), no. 3, 345–386. MR 1825963, DOI 10.1142/S0218216501000901
F. J.-B. J. Clauwens, The algebra of rack and quandle cohomology, arXiv:math/1004. 4423.
Pavel Etingof, Alexander Soloviev, and Robert Guralnick, Indecomposable set-theoretical solutions to the quantum Yang-Baxter equation on a set with a prime number of elements, J. Algebra 242 (2001), no. 2, 709–719. MR 1848966, DOI 10.1006/jabr.2001.8842
Roger Fenn, Colin Rourke, and Brian Sanderson, Trunks and classifying spaces, Appl. Categ. Structures 3 (1995), no. 4, 321–356. MR 1364012, DOI 10.1007/BF00872903
R. A. Litherland and Sam Nelson, The Betti numbers of some finite racks, J. Pure Appl. Algebra 178 (2003), no. 2, 187–202. MR 1952425, DOI 10.1016/S0022-4049(02)00211-6
Takuro Mochizuki, Some calculations of cohomology groups of finite Alexander quandles, J. Pure Appl. Algebra 179 (2003), no. 3, 287–330. MR 1960136, DOI 10.1016/S0022-4049(02)00323-7
Takuro Mochizuki, The 3-cocycles of the Alexander quandles $\Bbb F_q[T]/(T-\omega )$, Algebr. Geom. Topol. 5 (2005), 183–205. MR 2135551, DOI 10.2140/agt.2005.5.183
T. Nosaka, Quandle homotopy invariants of knotted surfaces, arXiv:math/1011. 6035.
M. Niebrzydowski and J. H. Przytycki, Homology of dihedral quandles, J. Pure Appl. Algebra 213 (2009), no. 5, 742–755. MR 2494367, DOI 10.1016/j.jpaa.2008.09.010
T. Ohtsuki, Problems on invariants of knots and 3-manifolds, Invariants of knots and 3-manifolds (Kyoto, 2001) Geom. Topol. Monogr., vol. 4, Geom. Topol. Publ., Coventry, 2002, pp. i–iv, 377–572. With an introduction by J. Roberts. MR 2065029, DOI 10.2140/gtm.2002.4