Abstract
We prove that the lower bounds for Betti numbers of the rack, quandle and degeneracy cohomology given in Carter et al. (J. Pure Appl. Algebra, 157 (2001) 135) are in fact equalities. We compute as well the Betti numbers of the twisted cohomology introduced in Carter et al. (Twisted quandle cohomology theory and cocycle knot invariants, math. GT/0108051). We also give a group-theoretical interpretation of the second cohomology group for racks.
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