On quandle homology groups of Alexander quandles of prime order
Author:
Takefumi Nosaka
Journal:
Trans. Amer. Math. Soc. 365 (2013), 34133436
MSC (2010):
Primary 20G10, 55N35, 58H10; Secondary 57Q45, 57M25, 55S20
Published electronically:
January 30, 2013
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Abstract 
References 
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Additional Information
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.
 [AS]
S. Asami, S. Satoh, An infinite family of noninvertible surfaces in space, Bull. London Math. Soc. 37 (2005), 285296. MR 2119028 (2005k:57044)
 [CJKLS]
J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford, M. Saito, Quandle cohomology and statesum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003) 39473989. MR 1990571 (2005b:57048)
 [CKS]
J. S. Carter, S. Kamada, M. Saito, Geometric interpretations of quandle homology, J. Knot Theory Ramifications 10 (2001) 345386. MR 1825963 (2002h:57009)
 [Cla]
F. J.B. J. Clauwens, The algebra of rack and quandle cohomology, arXiv:math/1004. 4423.
 [EGS]
P. Etingof, R. Guralnick, A. Soloviev, Indecomposable settheoretical solutions to the quantum YangBaxter equation on a set with a prime number of elements, J. Algebra 242 (2001) 709719. MR 1848966 (2002e:20049)
 [FRS]
R. Fenn, C. Rourke, B. Sanderson, Trunks and classifying spaces, Appl. Categ. Structures 3 (1995) 321356. MR 1364012 (96i:57023)
 [LN]
R. A. Litherland, S. Nelson, The Betti numbers of some finite racks, J. Pure Appl. Algebra 178, 2003, 187202. MR 1952425 (2004a:57006)
 [M1]
T. Mochizuki, Some calculations of cohomology groups of finite Alexander quandles, J. Pure Appl. Algebra 179 (2003) 287330. MR 1960136 (2004b:55013)
 [M2]
, The cocycles of the Alexander quandles , Algebraic and Geometric Topology. 5 (2005) 183205. MR 2135551 (2006e:57019)
 [No]
T. Nosaka, Quandle homotopy invariants of knotted surfaces, arXiv:math/1011. 6035.
 [NP]
M. Niebrzydowski, J. H. Przytycki. Homology of dihedral quandles, J. Pure Appl. Algebra 213 (2009) 742755. MR 2494367 (2010a:18014)
 [O]
T. Ohtsuki (ed.), Problems on invariants of knots and manifolds, Geom. Topol. Monogr., 4, Invariants of knots and 3manifolds (Kyoto, 2001), 377572, Geom. Topol. Publ., Coventry, 2002. MR 2065029 (2005c:57014)
 [AS]
 S. Asami, S. Satoh, An infinite family of noninvertible surfaces in space, Bull. London Math. Soc. 37 (2005), 285296. MR 2119028 (2005k:57044)
 [CJKLS]
 J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford, M. Saito, Quandle cohomology and statesum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003) 39473989. MR 1990571 (2005b:57048)
 [CKS]
 J. S. Carter, S. Kamada, M. Saito, Geometric interpretations of quandle homology, J. Knot Theory Ramifications 10 (2001) 345386. MR 1825963 (2002h:57009)
 [Cla]
 F. J.B. J. Clauwens, The algebra of rack and quandle cohomology, arXiv:math/1004. 4423.
 [EGS]
 P. Etingof, R. Guralnick, A. Soloviev, Indecomposable settheoretical solutions to the quantum YangBaxter equation on a set with a prime number of elements, J. Algebra 242 (2001) 709719. MR 1848966 (2002e:20049)
 [FRS]
 R. Fenn, C. Rourke, B. Sanderson, Trunks and classifying spaces, Appl. Categ. Structures 3 (1995) 321356. MR 1364012 (96i:57023)
 [LN]
 R. A. Litherland, S. Nelson, The Betti numbers of some finite racks, J. Pure Appl. Algebra 178, 2003, 187202. MR 1952425 (2004a:57006)
 [M1]
 T. Mochizuki, Some calculations of cohomology groups of finite Alexander quandles, J. Pure Appl. Algebra 179 (2003) 287330. MR 1960136 (2004b:55013)
 [M2]
 , The cocycles of the Alexander quandles , Algebraic and Geometric Topology. 5 (2005) 183205. MR 2135551 (2006e:57019)
 [No]
 T. Nosaka, Quandle homotopy invariants of knotted surfaces, arXiv:math/1011. 6035.
 [NP]
 M. Niebrzydowski, J. H. Przytycki. Homology of dihedral quandles, J. Pure Appl. Algebra 213 (2009) 742755. MR 2494367 (2010a:18014)
 [O]
 T. Ohtsuki (ed.), Problems on invariants of knots and manifolds, Geom. Topol. Monogr., 4, Invariants of knots and 3manifolds (Kyoto, 2001), 377572, Geom. Topol. Publ., Coventry, 2002. MR 2065029 (2005c:57014)
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Additional Information
Takefumi Nosaka
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Sakyoku, Kyoto, 6068502, Japan
Email:
nosaka@kurims.kyotou.ac.jp
DOI:
http://dx.doi.org/10.1090/S000299472013057546
PII:
S 00029947(2013)057546
Keywords:
Rack,
quandle,
homology,
cohomology,
knot
Received by editor(s):
November 17, 2009
Received by editor(s) in revised form:
April 1, 2011
Published electronically:
January 30, 2013
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
