Abstract
A partial classification of height atoms whose symmetry groups act transitively on its rings having the same color is obtained. Nine infinite series and 19 special cases are described.
Similar content being viewed by others
References
A. V. Bolsinov and A. T. Fomenko, Inteyrable Hamiltonian Systems. Geometry, Topology, Classification (Reg. Khaot. Dynam., Izhevsk, 1999; Chapman and Hall, CRC, Boca Raton. Fl., 2004).
V. O. Manturov, “Bifurcations, Atoms, and Knots,” Vestnik Mosk. Univ., Matem. Mekhan., No. 1, 3 (2000) [Moscow Univ. Math. Bull. 55 (1), 1 (2000)].
A. T. Fomenko, “The Topology of Surfaces of Constant Energy in Integrable Hamiltonian Systems, and Obstructions to Integrability,” Izvestiya Akad. Nauk SSSR, Ser. Matem. 50 (6), 1276 (1986) [Math, of the USSR-Izvestiya n29 (3), 629 (1987)].
A. T. Fomenko and H. Zieschang, “A Topological Invariant and a Criterion for the Equivalence of Integrable Hamiltonian Systems with Two Degrees of Freedom,” Izvestiya Akad. Nauk SSSR, Ser. Matem. 54 (3), 546 (1990) [Math, of the USSR-Izvestiya 36 (3), 567 (1991)].
A. T. Fomenko, E. A. Kudryavtseva, and I. M. Nikonov, “Maximally Symmetric Cell Decompositions of Surfaces and their Coverings,” Matem. Sbornik 199 (9), 3 (2008) [Sbornik: Math. 199 (9), 1263 ( 2008)].
E. A. Kudryavtseva, I. M. Nikonov, and A. T. Fomenko, “Symmetric and Irreducible Abstract Polyhedra,” in Sovremennye Problemy Matem,. Mekhan., ed. by A.T. Fomenko (Moscow State Univ., Moscow, 2009), pp. 58–97.
I. M. Nikonov, “Height Atoms whose Symmetry Groups Act Transitively on their Vertex Sets,” Vestnik Mosk. Univ., Matem. Mekhan., No. 6, 1 (2016) [Moscow Univ. Math. Bull. 71 (6), 233 (2016)]
I. M. Nikonov and N. V. Volchanetskii, “Maximally Symmetric Height Atoms,” Vestnik Mosk. Univ., Matem. Mekhan., No. 2, 3 (2013) [Moscow Univ. Math. Bull. 68 (2), 83 (2013)].
A. T. Fomenko and A. Yu. Konyaev, “New Approach to Symmetries and Singularities in Integrable Hamiltonian Systems,” Topol. and its Appl. 159, 1964 (2012).
A. T. Fomenko and E. A. Kudryavtseva, “Symmetries Groups of Nice Morse Functions on Surfaces,” Doklady Russ. Akad. Nauk, Ser. Matem. 446 (6), 615 (2012) [Doklady Math. 86 (2), 691 (2012)].
A. T. Fomenko and E. A. Kudryavtseva, “Each Finite Group is a Symmetry Group of Some Map (an “Atom“- Bifurcation),” Vestnik Mosk. Univ., Matem. Mekhan., No. 3, 21 (2013) [Moscow Univ. Math. Bull. 68 (3), 148 (2013)].
V. V. Vedyushkina (Fokicheva) and A. T. Fomenko, “Integrable Topological Billiards and Equivalent Dynamical Systems,” Izvestiya Russ. Akad. Nauk, Ser. Matem. 81 (4), 20 (2017) [Izvestiya: Math. 81 (4), 688 (2017)].
V. V. Vedyushkina (Fokicheva) and A. T. Fomenko, “Billiard Systems as the Models for the Rigid Body Dynamics,” in Studies in System,s, Decision and Control. Advances in Dynamical System,s and Control, ed. by V. A. Sadovnichiy and M. Z. Zgurovsky (Springer Int. Publ., Switzerland, 2016), Vol. 69, pp. 13–32.
V. V. Fokicheva and A. T. Fomenko, “Integrable Billiards Model Important Integrable Cases of Rigid Body Dynamics,” Doklady Russ. Akad. Nauk, Matem. 465 (2), 1 (2015) [Doklady. Math. 92 (3), 1 (2015)].
A. T. Fomenko and S. S. Nikolaenko, “The Chaplygin Case in Dynamics of a Rigid Body in Fluid is Orbitally Equivalent to the Euler Case in Rigid Body Dynamics and to the Jacobi Problem about Geodesies on the Ellipsoid,” J. Geom. and Phys. 87, 115 (2015).
A. T. Fomenko and E. O. Kantonistova, “Topological Classification of Geodesic Flows on Revolution 2-Surfaces with Potential,” in Continuous and Distributed System,s II. Theory and Applications, ed. by V. A. Sadovnichiy and M. Z. Zgurovsky (Springer Int. Publ., Cham, Heidelberg, N.Y., Dordrecht, L., 2015), pp. 11–17.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.A. Trifonova. 2018. published in Vestnik Moskovskogo Universiteta. Matematika. Mekhanika. 2018. Vol. 73, No. 2, pp. 33–41.
About this article
Cite this article
Trifonova, V.A. Partially Symmetric Height Atoms. Moscow Univ. Math. Bull. 73, 71–78 (2018). https://doi.org/10.3103/S0027132218020043
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0027132218020043