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Plans’ periodicity theorem for Jacobian of circulant graphs. (English. Russian original) Zbl 1477.57007

Dokl. Math. 103, No. 3, 139-142 (2021); translation from Dokl. Ross. Akad. Nauk, Mat. Inform. Protsessy Upr. 498, 51-54 (2021).
Summary: Plans’ theorem [A. Plans, Rev. Acad. Ci. Madrid 47, 161–193 (1953; Zbl 0051.14603)] 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 \(n\).

MSC:

57K10 Knot theory
57M15 Relations of low-dimensional topology with graph theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)

Citations:

Zbl 0051.14603
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References:

[1] Goel, G.; Perkinson, D., Linear Algebra Appl., 567, 138-142 (2019) · Zbl 1411.05161 · doi:10.1016/j.laa.2019.01.009
[2] L. A. Grunwald and I. A. Mednykh, “On complexity and Jacobian of cone over a graph,” Preprint (2020) arXiv:2004.07452 [math.CO].
[3] Mednykh, A. D.; Mednykh, I. A., Dokl. Math., 94, 445-449 (2016) · Zbl 1350.05061 · doi:10.1134/S106456241604027X
[4] Kawauchi, A., A Survey of Knot Theory (1996), Basel: Birkhäuser, Basel · Zbl 0861.57001
[5] Kwon, Y. S.; Mednykh, A. D.; Mednykh, I. A., Linear Algebra Appl., 529, 355-373 (2017) · Zbl 1365.05135 · doi:10.1016/j.laa.2017.04.032
[6] Plans, A., Rev. Real Acad. Cienc. Exact. Fis. Natur. Madrid, 47, 161-193 (1953) · Zbl 0051.14603
[7] Gordon, C. McA., Bull. Am. Math. Soc., 77, 85-87 (1971) · Zbl 0206.52301 · doi:10.1090/S0002-9904-1971-12611-3
[8] W. H. Stevens, PhD Thesis (Louisiana State University, 1996). https://digitalcommons.lsu.edu/gradschool_disstheses/6282.
[9] Sakuma, M., Can. J. Math., 47, 201-224 (1995) · Zbl 0839.57001 · doi:10.4153/CJM-1995-010-2
[10] N. Neumӓrker, PhD Thesis (Univ. Bielefeld, Bielefeld, 2012).
[11] Carmichael, R. D., Quart. J. Pure Appl. Math., 48, 343-372 (1920)
[12] Ward, M., Trans. Am. Math. Soc., 35, 600-628 (1933) · Zbl 0007.24901 · doi:10.1090/S0002-9947-1933-1501705-4
[13] Kwon, Y. S.; Mednykh, A. D.; Mednykh, I. A., Dokl. Math., 99, 286-289 (2019) · Zbl 1426.05081 · doi:10.1134/S1064562419030141
[14] Fox, R. H., Ann. Math., 71, 187-196 (1960) · Zbl 0122.41801 · doi:10.2307/1969886
[15] Helling, H.; Kim, A. C.; Mennicke, J. L., J. Lie Theory, 8, 1-23 (1998) · Zbl 0896.20026
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