Geodesic
Plans' periodicity theorem for Jacobian of circulant graphs
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 498 (2021), pp. 51-54

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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 $n$.
Keywords: Alexander polynomial, knot, knot branched covering, circulant graph, critical group, cyclic covering, homology group. 
@article{DANMA_2021_498_a9,
     author = {A. D. Mednykh and I. A. Mednykh},
     title = {Plans' periodicity theorem for {Jacobian} of circulant graphs},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {51--54},
     publisher = {mathdoc},
     volume = {498},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2021_498_a9/}
}
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A. D. Mednykh; I. A. Mednykh. Plans' periodicity theorem for Jacobian of circulant graphs. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 498 (2021), pp. 51-54. http://geodesic.mathdoc.fr/item/DANMA_2021_498_a9/