On a fundamental polyhedron of a hyperbolic cone-manifold

Authors

Lilya Grunwald, Sobolev Institute of Mathematics, Novosibirsk, Russian FederationFollow
Aydos Qutbaev, Novosibirsk State University, Novosibirsk, Russian FederationFollow

Abstract

In this paper we propose to consider two different solutions having a common problem of establishing hyperbolic structure on a 3-manifold. Under the consideration will be a cone 3-manifold with underlying space as a 3-sphere and a singular set nested in it. Furthermore, this paper is divided into two cases: a singular set as the 3₁ knot with a bridge and a singular set as the 6₁³ link. The hyperbolic space $H³ for the analytical examination in the first case will be a hyperboloid model, in the second using the upper-half space model. To show that the cone-manifold admits a hyperbolic structure, its fundamental set was constructed in the space of each case and the conditions of its existence were provided. In addition, for the first case we will present the analytical formula to calculate the hyperbolic volume of its manifold.

First Page

213

Last Page

228

References

1. Mednykh A. and Rasskazov A. Volumes and Degeneration of Cone-structures on the Figure-eight Knot. Tokyo J.Math. Vol. 29 (2006).

2. Mednykh A.D. Volumes of two-bridge cone manifolds in spaces of constant curvature, Transformation Groups. Vol. 26, Springer, pp. 601-629 (2020).

3. Mednykh A.D. and Sokolova D.Yu. The existence of an Euclidian structure on the Figure-eight Knot. Yak. Math. Journal (2015).

4. Abrosimov N., Vuong B. On the volume of hyperbolic polyhedra. Journal of Knot Theory and Its Ramifications (2021).

5. Arkadiusz Jadczyk. Quantum Fractals: From Heisenberg’s Uncertainty to Barnsley’s Fractality. World Scientific Publishing Co., Singapore (2014).

6. Brin M.G., Jones G.A., Singerman D. Commentary on Robert Riley’s article “A personal account of the discovery of hyperbolic structures on some knot complements”. Expo. Math. 31, No. 2, 99–103 (2013).

7. Cooper D., Hodgson C.D., Kerckhoff S.P. Three-dimensional Orbifolds and Cone-manifolds. MSJ Memoirs, Vol. 5, Mathematical Society of Japan, Tokyo (2000).

8. Derevnin D., Mednykh A., Mulazzani M. Volumes for twist link cone-manifolds. arXiv:math/0307193v2 [math.GT], 4 Mar (2004).

9. Ji-Young Ham, Joongul Lee, Alexander Mednykh and Aleksey Rasskazov. On the volume and Chern-Simons invariant for the 2-bridge knot orbifolds.

10. James W. Cannon, William J. Floyd, Richard Kenyon, and Walter R. Parry. Hyperbolic Geometry. Flavours of Geometry. MSRI Publications, Vol. 31 (1997).

11. Purcell Jessica S. Hyperbolic Knot theory. http://arxiv.org/abs/2002.12652v1.

12. Kellerhals R. On the volume of hyperbolic polyhedra. Math. Ann., Vol. 285, P. 541-569 (1989).

13. Maxwell J.C. A treatise on electricity and magnetism. Oxford (1883).

14. Kauffman L. New Invariants in the Theory of Knots. Amer. Math. Monthly, Vol. 95, No. 3, 195-242. Kauffman L. Knots and physics. University of Illinois, Chicago, USA (1991).

15. Marc Culler, Nathan M. Dunfield, Matthias Goerner, and Jeffrey R. Weeks. SnapPy, a computer program for studying the geometry and topology of 3-manifolds, Available at http://snappy.computop.org (2016).

16. Perelman G. The entropy formula for the Ricci flow and its geometric applications. math.DG/0211159

17. Perelman G. Ricci flow with surgery on three-manifolds. math.DG/0303109

18. Perelman G. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. math.DG/0307245

19. Riley R. Seven excellent knots in: Low-Dimension Topology (Bangor 1979), London Mathematical Society Lecture Note Series, Vol. 48, Cambridge University Press, Cambridge, pp. 81–151 (1982).

20. Riley R. A personal account of the discovery of hyperbolic structure on some knot complements. Expo. Math. 31, No. 2, 104–115 (2013).

21. Rolfsen D. Knots and Links. AMS Chelsea, Providence, third edition (2003).

22. Riley R. An elliptical path from parabolic representations to hyperbolic structure. Topology of Low–Dimension manifolds, LNM, 722, Springer–Verlag, P. 99–133 (1979).

23. Thurston W. The geometry and topology of 3-manifold. Lecture notes. : Princeton Univ. (1980).

24. Fenchel W. Elementary Geometry in Hyperbolic Space. De Gruyter Studies in Mathematics, Vol. 11, Walter de Gruyter, Berlin-New York (1989).

25. Hodgson C.D. Degeneration and regeneration of geometric structures on 3- manifolds. Ph.D. thesis, Princeton Univ. (1986).

26. Ham J.-Y., Mednykh A., Petrov V. Trigonometric identities and volumes of the hyperbolic twist knot cone-manifolds. J. Knot Theory Ramifications 23, No. 12, 1450064 (2014).

26. P. de la Harpe, M. Kervaire and C. Weber. On the Jones Polynomial. L’Enseignement Mathematique, Vol. 32, pp. 271-335 (1986).

27. Fenchel W. Elementary Geometry in Hyperbolic Space. De Gruyter Studies in Mathematics, Vol. 11, Walter de Gruyter, Berlin-New York (1989).

Recommended Citation

Grunwald, Lilya and Qutbaev, Aydos (2022) "On a fundamental polyhedron of a hyperbolic cone-manifold," Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 5: Iss. 4, Article 1.
DOI: https://doi.org/10.56017/2181-1318.1232