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

AGT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to