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Cyclic branched coverings of knots and quandle homology. (English) Zbl 1261.57015

In the article under review, the author applies a result of joint work with A. Inoue [Quandle homology and complex volume, preprint (2010); arXiv:1012.2923v1]. For any integer \(p>2\), let \(R_p\) denote the cyclic group \({\mathbb Z}/p\) with quandle operation defined by \(x*y=2y-x\) \(\mod\) \(p\). Actually this operation satisfies the quandle axioms, and \(R_p\) is called the dihedral quandle. The author constructs quandle cocycles of \(R_p\) from group cocycles of \(G={\mathbb Z}/p\), and proposes a general construction of quandle cocycles from group cocycles. The author proves that a group 3-cocycle of \({\mathbb Z}/p\) gives rise to a nontrivial quandle 3-cocycle of \(R_p\). When \(p\) is an odd prime, since \(\dim_{{\mathbb F}_{p}}H^{3}_{Q}(R_{p}; {\mathbb F}_{p})=1\), the author’s 3-cocycle is a constant multiple of the Mochizuki 3-cocycle up to coboundary. Let \(X\) be a quandle and \(D\) a diagram of a knot \(K\). For a shadow coloring \(\varphi=(S,R)\) of \(D\) whose arcs and regions are colored by \(X\), the author proves that the cycle \(c (\varphi)\) associated with \(\varphi=(S, R)\) of \(K\) gives rise to a group cycle represented by a cyclic branched covering along \(K\) and the representation induced from the arc coloring \(S\).

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
57M10 Covering spaces and low-dimensional topology
57M12 Low-dimensional topology of special (e.g., branched) coverings
57M25 Knots and links in the \(3\)-sphere (MSC2010)