Skip to main content
SpringerLink
Log in

The volume conjecture for polyhedra implies the Stoker conjecture

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We show that the Volume Conjecture for polyhedra implies a weak version of the Stoker Conjecture; in turn we prove that this weak version of the Stoker conjecture implies the Stoker conjecture. The main tool used is an extension of a result of Montcouquiol and Weiss, saying that dihedral angles are local coordinates for compact polyhedra with angles \(\le \pi \).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andreev, E.M.: On convex polyhedra in Lobachevskii spaces. Mat. Sb. 123(3), 445–478 (1970)

    MATH  Google Scholar 

  2. Bao, X., Bonahon, F.: Hyperideal polyhedra in hyperbolic 3-space. Bull. Soc. Math. de France 130(3), 457–491 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Belletti, G., Detcherry, R., Kalfagianni, E., Yang, T.: Growth of quantum 6\(j\)-symbols and applications to the volume conjecture. J. Differ. Geom. 120(2), 199–229 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  4. Belletti, G.: A maximum volume conjecture for hyperbolic polyhedra. arXiv:2002.01904, (2020)

  5. Belletti, G.: The maximum volume of hyperbolic polyhedra. Trans. Am. Math. Soc. 374(2), 1125–1153 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. Costantino, F., Guéritaud, F., van der Veen, R.: On the volume conjecture for polyhedra. Geom. Dedicata 179(1), 385–409 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, Q., Murakami, J.: Asymptotics of quantum 6\(j\)-Symbols. arXiv preprint math.GT/1706.04887

  8. Costantino, F.: \(6j\)-symbols, hyperbolic structures and the volume conjecture. Geom. Topol. 11, 1831–1854 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, Q., Yang, T.: Volume conjectures for the Reshetikhin-Turaev and the Turaev-Viro invariants. Quantum Topol. 9(3), 419–460 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fleischner, H.: The uniquely embeddable planar graphs. Discrete Math. 4(4), 347–358 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kashaev, R.: The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39(3), 269–275 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kolpakov, A., Murakami, J.: Combinatorial decompositions, Kirillov–Reshetikhin invariants, and the volume conjecture for hyperbolic polyhedra. Exp. Math. 27(2), 193–207 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Milnor, J.W.: Collected papers. 1. Geometry. Publish or Perish, (1994)

  14. Murakami, H., Murakami, J.: The colored jones polynomials and the simplicial volume of a knot. Acta Math. 186(1), 85–104 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Montcouquiol, G.: Deformations of hyperbolic convex polyhedra and cone-3-manifolds. Geom. Dedicata 166(1), 163–183 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Montcouquiol, G., Weiß, H.: Complex twist flows on surface group representations and the local shape of the deformation space of hyperbolic cone-3-manifolds. Geom. Topol. 17(1), 369–412 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ohtsuki, T.: On the asymptotic expansion of the Kashaev invariant of the \(5_2\) knot. Quantum Topol. 7(4), 669–735 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ohtsuki, T.: On the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings. Int. J. Math. 28(13), 1750096 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ohtsuki, T.: On the asymptotic expansion of the quantum \(SU(2)\) invariant at \(q=\exp (4\pi \sqrt{-1}/N)\) for closed hyperbolic \(3\)-manifolds obtained by integral surgery along the figure-eight knot. Algebr. Geom. Topol. 18(7), 4187–4274 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ohtsuki, T., Yokota, Y.: On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 165, pages 287–339. Cambridge University Press, (2018)

  21. Rivin, I., Hodgson, C.: A characterization of compact convex polyhedra in hyperbolic 3-space. Invent. Math. 111(1), 77–111 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rivin, I.: A characterization of ideal polyhedra in hyperbolic 3-space. Ann. Math. 143, 51–70 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Steinitz, E.: Polyeder und raumeinteilungen. Encyk. der Math. Wiss. 12, 38–43 (1922)

    Google Scholar 

  24. Stoker, J.: Geometrical problems concerning polyhedra in the large. Commun. Pure Appl. Math. 21(2), 119–168 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  25. Weiss, H.: The deformation theory of hyperbolic cone-3-manifolds with cone-angles less than 2\(\pi \). Geom. Topol. 17(1), 329–367 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yokota, Y.: Topological invariants of graphs in 3-space. Topology 35(1), 77–87 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université Paris-Saclay, Orsay, France

    Giulio Belletti

Authors
  1. Giulio Belletti
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

GB is the sole author.

Corresponding author

Correspondence to Giulio Belletti.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belletti, G. The volume conjecture for polyhedra implies the Stoker conjecture. Geom Dedicata 217, 77 (2023). https://doi.org/10.1007/s10711-023-00813-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10711-023-00813-y

Search

Navigation