Cyclic branched coverings of knots and quandle homology

TL;DR

This paper constructs quandle cocycles from group cocycles, especially for dihedral quandles, and relates them to cyclic branched coverings of knots, revealing new connections between quandle homology and knot invariants.

Contribution

It introduces a method to derive quandle cocycles from group cocycles, particularly for dihedral quandles, and links these to cyclic branched coverings of knots.

Findings

A construction of quandle cocycles from group cocycles for dihedral quandles.

Identification of the Mochizuki 3-cocycle as a special case.

Establishment of a correspondence between quandle cycles and cyclic branched coverings.

Abstract

We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since dim_{F_p} H_Q^3(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.