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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - A. D. Mednykh
AU  - I. A. Mednykh
TI  - Plans' periodicity theorem for Jacobian of circulant graphs
JO  - Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
PY  - 2021
SP  - 51
EP  - 54
VL  - 498
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DANMA_2021_498_a9/
LA  - ru
ID  - DANMA_2021_498_a9
ER  - 
%0 Journal Article
%A A. D. Mednykh
%A I. A. Mednykh
%T Plans' periodicity theorem for Jacobian of circulant graphs
%J Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
%D 2021
%P 51-54
%V 498
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DANMA_2021_498_a9/
%G ru
%F DANMA_2021_498_a9
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/

[1] Goel G., Perkinson D., Linear Algebra Appl., 567 (2019), 138–142 | DOI | MR | Zbl

[2] Grunwald L.A., Mednykh I.A., On complexity and Jacobian of cone over a graph, 2020, arXiv: 2004.07452 [math.CO]

[3] Mednykh A.D., Mednykh I.A., DAN, 469:5 (2016), 539–543 | Zbl

[4] Kawauchi A., A survey of knot theory, Birkhauser Verlag, Basel, 1996 | MR | Zbl

[5] Kwon Y.S., Mednykh A.D., Mednykh I.A., Linear Algebra Appl., 529 (2017), 355–373 | DOI | MR | Zbl

[6] Plans A., Rev. Real Acad. Cienc. Exact. Fis. Natur. Madrid, 47 (1953), 161–193 | Zbl

[7] Gordon C.McA., Bull. Amer. Math. Soc., 77 (1971), 85–87 | DOI | MR | Zbl

[8] Stevens W.H., On the Homology of Branched Cyclic Covers of Knots, PhD thesis, Louisiana State University, 1996 https://digitalcommons.lsu.edu/gradschool_disstheses/6282

[9] Sakuma M., Can. J. Math., 47:1 (1995), 201–224 | DOI | MR | Zbl

[10] Neumärker N., The Arithmetic Structure of Discrete Dynamical Systems on the Torus, PhD thesis, Univ. Bielefeld, 2012

[11] Carmichael R.D., Quart. J. Pure Appl. Math., 48 (1920), 343–372

[12] Ward M., Trans. Amer. Math. Soc., 35 (1933), 600–628 | DOI | MR | Zbl

[13] Kvon I.S., Mednykh A.D., Mednykh I.A., DAN, 486:4 (2019), 411–415

[14] Fox R.H., Ann. Math., 71:1 (1960), 187–196 | DOI | MR | Zbl

[15] Helling H., Kim A.C., Mennicke J.L., J. Lie Theory, 8:1 (1998), 1–23 | MR | Zbl