Cyclic branched coverings of knots and quandle homology

Yuichi Kabaya

Vol. 259 (2012), No. 2, 315–347

DOI: 10.2140/pjm.2012.259.315

Abstract

We give a construction of quandle cocycles from group cocycles, especially, for any integer p ≥ 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group ℤ∕p. We show that a group 3-cocycle of ℤ∕p gives rise to a nontrivial quandle 3-cocycle of R_p. When p is an odd prime, since dim_𝔽pH_Q³(R_p; 𝔽_p) = 1, our 3-cocycle is a constant multiple of the Mochizuki 3-cocycle up to coboundary. Dually, we construct a group cycle represented by a cyclic branched covering branched along a knot K from the quandle cycle associated with a colored diagram of K.

Keywords

quandle homology, group homology, cyclic branched covering

Mathematical Subject Classification 2010

Primary: 57M05, 57M10, 57M12, 57M25, 57M27

Milestones

Received: 9 November 2011

Revised: 13 April 2012

Accepted: 16 April 2012

Published: 3 October 2012