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Twisted quandle homology theory and cocycle knot invariants

J Scott Carter, Mohamed Elhamdadi and Masahico Saito

Algebraic & Geometric Topology 2 (2002) 95–135

arXiv: math.GT/0108051


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.

quandle homology, cohomology extensions, dihedral quandles, Alexander numberings, cocycle knot invariants
Mathematical Subject Classification 2000
Primary: 57N27, 57N99
Secondary: 57M25, 57Q45, 57T99
Forward citations
Received: 27 September 2001
Accepted: 8 February 2002
Published: 14 February 2002
J Scott Carter
University of South Alabama
Mobile AL 36688
Mohamed Elhamdadi
University of South Florida
Tampa FL 33620
Masahico Saito
University of South Florida
Tampa FL 33620