Algebraic and Geometric Topology 2 (2002),
paper no. 6, pages 95-135.
Twisted quandle homology theory and cocycle knot invariants
J. Scott Carter, Mohamed Elhamdadi, Masahico Saito
Abstract.
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.
Keywords.
Quandle homology, cohomology extensions, dihedral quandles, Alexander numberings, cocycle knot invariants
AMS subject classification.
Primary: 57N27, 57N99.
Secondary: 57M25, 57Q45, 57T99.
DOI: 10.2140/agt.2002.2.95
E-print: arXiv:math.GT/0108051
Submitted: 27 September 2001.
Accepted: 8 February 2002.
Published: 14 February 2002.
Notes on file formats
J. Scott Carter, Mohamed Elhamdadi, Masahico Saito
University of South Alabama, Mobile, AL 36688, USA
University of South Florida, Tampa, FL 33620, USA
University of South Florida, Tampa, FL 33620, USA
Email: carter@mathstat.usouthal.edu, emohamed@math.usf.edu, saito@math.usf.edu
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