Plans’ Periodicity Theorem for Jacobian of Circulant Graphs

Plans’ theorem states that, for odd n, the first homology group of the n-fold cyclic covering of the three-dimensional sphere branched over a knot is the direct product of two copies of an Abelian group. A similar statement holds for even n. In this case, one has to factorize the homology group of n-fold covering by the homology group of two-fold covering of the knot. The aim of this paper is to establish similar results for Jacobians (critical group) of a circulant graph. Moreover, it is shown that the Jacobian group of a circulant graph on n vertices reduced modulo a given finite Abelian group is a periodic function of n.