Abstract
For more than almost 200 years the Möbius strip and its “mysterious” property attracts the attention of mathematicians. After a “complete cut” of this surface, one object appears, but already with a fourfold twist. The generalization of this phenomenon to figures of a more complex configuration led to an “unexpected” result: after the cut of the generalized Möbius-Listing body, more than two geometric shapes may appear. In this paper, we consider all possible cases of a complete cut of the generalized Möbius-Listing body with a regular hexagon as radial section. In early works, together with different colleagues, on the basis of importance, they separately examined the case of Möbius-Listing’s bodies with a radial section of regular 3, 4 and 5 angular figures. Also, cases of similar bodies with a radial section of convex regular two and three angular figures were considered separately. One possible application of these results is assumed in the description of the properties of the middle surfaces in the theory of elastic shells [14] (Vekua, Shell Theory: General Methods of Construction. Pitman Advanced Publishing Program, Boston, p. 287, 1985).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Caratelli, D., Gielis, J., Ricci, P.E., Tavkhelidze, I.: The Dirichlet Problem for the Laplace Equation in Supershaped Annuli. Bound. Value Probl (Springer Open Journal) 2013, 113 (2013). Ded. to Prof Hari M. Srivastava
Doll, H., Hoste, J.: A tabulation of oriented links. Math. Comput. 57, 747–761 (1991)
Gielis, J., Caratelli, D., Fougerolle, Y., Ricci, P.E., Tavkhelidze, I., Gerats, T.: Universal natural shapes: from unifying shape description to simple methodsfor shape analysis and boundary value problems. PlosONE-D-11-01115R2 27, IX, 1–18 (2012). https://doi.org/10.1371/journal.pone.0029324
Gielis, J.: The Geometrical Beauty of Plants, pp. 1–229. Atlantis Press (2017)
Gray, A., Albena, E., Salamon, S.: Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd edn. J. Capman and Hall/CRC, Boca Raton
Kupenberg, G.: Quadrisecants of knots and links. J. Knot Theory Ramif. 3, 41–50 (1994)
Matsuura, M.: Gielis superformula and regular polygons. J. Geometry 106(2), 383–403 (2015)
Sosinsky, A.: Knots Mathematics with a Twist, pp. 1–147. Harvard University Press, Cambridge (2002)
Tavkhelidze, I., Caratelli, D., Gielis, J., Ricci, P.E., Rogava, M., Transirico, M.: On a Geometric Model of Bodies with “Complex" Configuration and Some Movements - Modeling in Mathematics- Chapter 10. Atlantis Transactions in Geometry, vol. 2, pp. 129–158. Springer, Berlin (2017). https://doi.org/10.2991/978-94-6239-261-810
Tavkhelidze, I., Ricci, P.E.: Rendiconti Accademia Nazionale dell Scienze detta dei XL Memorie di Matematica a Applicazioni, 1240 vol. XXX, fasc. 1, 191–212 (2006)
Tavkhelidze, I., Ricci, P.E.: Some Properties of “Bulky” Links, Generated by Generalised Möbius-Listing’s Bodies - Modeling in Mathematics- Chapter 11. Atlantis Transactions in Geometry, vol. 2, pp. 158–185. Springer, Berlin (2017). https://doi.org/10.2991/978-94-6239-261-811
Tavkhelidze, I.: About connection of the generalized Möbius-Listing’s surfaces with sets of ribbon knots and links. In: Proceedings of Ukrainian Mathematical Congress, S.2 Topology and Geometry, Kiev - 2011, pp. 177–190 (2011)
Tavkhelidze, I., Cassisa, C., Gielis, J., Ricci, P.E.: About “Bulky” Links, Generated by Generalized Möbius-Listing’s bodies. Rendiconti Lincei Mat. Appl. 24, 11–38 (2013)
Vekua, I.: Shell Theory: General Methods of Construction, p. 287. Pitman Advanced Publishing Program, Boston (1985)
Weisstein, E.W.: The CRC Concise Encyclopedia of Mathematics, 2nd edn. Chapman & Hall/CRC, Boca Raton (2003)
Acknowledgements
The authors are very grateful to Johan Gielis for valuable comments. Also, the authors are grateful to Paolo Emilio Ricci and Diego Caratelly for valuable discussions. Some part of project has been fulfilled by a financial support of Shota Rustaveli National Science Foundation (Grant SRNSF/FR/358/5-109/14), also Some details of the article were finalized and added during I. Tavkhelidze’s visit to Portugal to the conference ICDDEA-2017.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Pinelas, S., Tavkhelidze, I. (2018). Analytic Representation of Generalized Möbius-Listing’s Bodies and Classification of Links Appearing After Their Cut. In: Pinelas, S., Caraballo, T., Kloeden, P., Graef, J. (eds) Differential and Difference Equations with Applications. ICDDEA 2017. Springer Proceedings in Mathematics & Statistics, vol 230. Springer, Cham. https://doi.org/10.1007/978-3-319-75647-9_38
Download citation
DOI: https://doi.org/10.1007/978-3-319-75647-9_38
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-75646-2
Online ISBN: 978-3-319-75647-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)