TY - JOUR

T1 - Plans’ Periodicity Theorem for Jacobian of Circulant Graphs

AU - Mednykh, A. D.

AU - Mednykh, I. A.

N1 - Funding Information: This work was supported by the Mathematical Center in Akademgorodok, agreement no. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation. Publisher Copyright: © 2021 Pleiades Publishing, Ltd.

PY - 2021/5

Y1 - 2021/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85114041719&partnerID=8YFLogxK

U2 - 10.1134/S1064562421030121

DO - 10.1134/S1064562421030121

M3 - Article

AN - SCOPUS:85114041719

VL - 103

SP - 139

EP - 142

JO - Doklady Mathematics

JF - Doklady Mathematics

SN - 1064-5624

IS - 3

ER -