Advanced search
Publication Cover
Experimental Mathematics Volume 19, 2010 - Issue 2
Submit an article Journal homepage
40
Views
4
CrossRef citations to date
0
Altmetric
Original Article

Hyperbolic Graphs of Small Complexity

Damian Heard Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010, Australia
,
Craig Hodgson Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010, Australia
,
Bruno Martelli Dipartimento di Matematica "Tonelli,", Largo Pontecorvo 5, 56127 Pisa, Italy
&
Carlo Petronio Dipartimento di Matematica Applicata "Dini,", Via Buonarroti 1/C, 56127 Pisa, Italy
Pages 211-236 | Published online: 30 Jan 2011

  • M. Boileau, B. Leeb, and J. Porti. "Geometrization of 3-Dimensional Orbifolds." Ann. of Math. 162 (2005), 195–290.
  • P. J. Callahan, M. V. Hildebrand, and J. R. Weeks. "A Census of Cusped Hyperbolic 3-Manifolds." Math. Comp. 68 (1999), 321–332.
  • C. Cao and G. Meyerhoff. "The Orientable Cusped Hyperbolic 3-Manifolds of Minimum Volume." Invent. Math. 146 (2001), 451–478.
  • M. Chiodo, D. Heard, C. Hodgson, J. Saunderson, and N. Sheridan. "Enumeration and Classification of Knotted Graphs in S3." In preparation, 2010.
  • J. Conway. "An Enumeration of Knots and Links, and Some of Their Algebraic Properties." In Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), pp. 329–358. Oxford: Pergamon, 1970.
  • D. Cooper, C. Hodgson, and S. Kerckhoff. "Three-Dimensional Orbifolds and Cone Manifolds, Mathematical Society of Japan Memoirs 5. Tokyo, 2000.
  • D. Coulson, O. Goodman, C. Hodgson, and W. Neumann. "Computing Arithmetic Invariants of 3-Manifolds." Experiment. Math. 9 (2000), 127–152.
  • D. B. A. Epstein and R. C. Penner. "Euclidean Decompositions of Noncompact Hyperbolic Manifolds." J. Differential Geom. 27 (1988), 67–80.
  • E. Flapan. When Topology Meets Chemistry. A topological Look at Molecular Chirality. Cambridge, UK: Cambridge University Press, 2000.
  • R. Frigerio and C. Petronio. "Construction and Recognition of Hyperbolic 3-Manifolds with Geodesic Boundary." Trans. Amer. Math. Soc. 356 (2004), 3243–3282.
  • R. Frigerio, B. Martelli, and C. Petronio. "Small Hyperbolic 3-Manifolds with Geodesic Boundary." Experiment. Math. 13 (2004), 171–184.
  • M. Fujii. "Hyperbolic 3-Manifolds with Totally Geodesic Boundary Which Are Decomposed into Hyperbolic Truncated Tetrahedra." Tokyo J. Math. 13 (1990), 353–373.
  • D. Gabai, R. Meyerhoff, and P. Milley. "Mom Technology and Volumes of Hyperbolic 3-Manifolds." arXiv:math.GT/0606072, 2006.
  • D. Gabai, R. Meyerhoff, and P. Milley. "Minimum Volume Cusped Hyperbolic Three-Manifolds." J. Amer. Math. Soc. 22 (2009), 1157–1215.
  • F. W. Gehring and G. J. Martin. "Minimal Co-volume Hyperbolic Lattices I: The Spherical Points of a Kleinian Group." Ann. of Math. 170 (2009), 123–161.
  • D. Heard. "Computation of Hyperbolic Structures on 3-Dimensional Orbifolds." PhD thesis, University of Melbourne, 2005. Available online ( www.ms.unimelb.edu.au/∼snap/DHeard-PhD.pdf
  • J. Hoste, M. Thistlethwaite, and J. Weeks. "The First 1,701,936 Knots." Math. Intelligencer 20:4, (1998), 33–48.
  • W. Jaco and J. H. Rubinstein. "Layered Triangulations of 3-Manifolds." arXiv:math.GT/0603601, 2006.
  • M. Kapovich. Hyperbolic Manifolds and Discrete Groups, Progress in Mathematics 183. Boston: Birkhäuser, 2001.
  • S. Kojima. "Polyhedral Decomposition of Hyperbolic Manifolds with Boundary." Proc. Work. Pure Math. 10 (1990), 37–57.
  • S. Kojima. "Polyhedral Decomposition of Hyperbolic 3-Manifolds with Totally Geodesic Boundary." In Aspects of Low-Dimensional Manifolds, Adv. Stud. Pure Math. 20, pp. 93–112. Tokyo: Kinokuniya, 1992.
  • S. Kojima and Y. Miyamoto. "The Smallest Hyperbolic 3-Manifolds with Totally Geodesic Boundary." J. Differential Geom. 34 (1991), 175– 192.
  • R. Litherland. "A Table of All Prime Theta-Curves in S3 up to 7 Crossings." Personal communication, 1989.
  • C. Maclachlan and A. Reid. The Arithmetic of Hyperbolic 3-Manifolds. New York: Springer-Verlag, 2003.
  • T. H. Marshall and G. J. Martin. "Minimal Co-volume Hyperbolic Lattices, II: Simple Torsion in Kleinian Groups." Preprint, 2008.
  • B. Martelli and C. Petronio. "3-Manifolds Having Complexity at Most 9." Experiment. Math. 10 (2001), 207–237.
  • B. Martelli and C. Petronio. "Complexity of Geometric 3-Manifolds." Geom. Dedicata 108 (2004), 15–69.
  • S. V. Matveev. "Transformations of Special Spines, and the Zeeman Conjecture" (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 1104–1116, 1119. English translation: Math. USSR-Izv. 31 (1988), 423–434.
  • S. V. Matveev. "Complexity Theory of Three-Dimensional Manifolds." Acta Appl. Math. 19 (1990), 101– 130.
  • S. V. Matveev. "Tabulation of Three-Dimensional Manifolds." Russ. Math. Surv. 60 (2005), 673–698.
  • S. V. Matveev. Algorithmic Topology and Classification of 3-Manifolds, Algorithms and Computation in Mathematics 9. Berlin: Springer, 2003.
  • Y. Miyamoto. "Volumes of Hyperbolic Manifolds with Geodesic Boundary." Topology 33 (1994), 613– 629.
  • J. Morgan. "On Thurston's Uniformization Theorem for Three-Dimensional Manifolds." In The Smith Conjecture (New York, 1979), Pure Appl. Math. 112, pp. 37–125. Orlando: Academic Press, 1984.
  • H. Moriuchi. "An Enumeration of Theta-Curves with up to Seven Crossings." In Proceedings of the East Asian School of Knots, Links, and Related Topics, 2004, Seoul, Korea, pp. 171–185. Available online (knot.kaist.ac.kr/2004/proceedings/MORIUCHI.pdf), 2004.
  • H. Moriuchi. "A Table of Handcuff Graphs with up to Seven Crossings." In Proceedings of the International Workshop on Knot Theory for Scientific Objects, 2006, Osaka, pp. 171–185. Available online ( www.omup.jp/modules/papers/knot/chap15.pdf
  • C. Petronio. "Complexity of 3-Orbifolds." Topology Appl. 153 (2006), 1658–1681.
  • C. Petronio. "Spherical Splitting of 3-Orbifolds." Math. Proc. Cambridge Philos. Soc. 142 (2007), 269–287.
  • R. Piergallini. "Standard Moves for Standard Polyhedra and Spines." Rend. Circ. Mat. Palermo (2) Suppl. 18 (1988), 391–414.
  • D. Rolfsen. Knots and Links. Berkeley: Publish or Perish, 1976.
  • C. Rourke and B. Sanderson. Introduction to Piecewise Linear Topology, Ergebn. der Math. 69. New York: Springer, 1972.
  • J. Simon, "A Topological Approach to the Stereochemistry of Nonrigid Molecules." In Graph Theory and Topology in Chemistry (Athens, Ga., 1987), Stud. Phys. Theoret. Chem. 51, pp. 43–75. Amsterdam: Elsevier, 1987.
  • W. P. Thurston. "Geometry and Topology of 3-Manifolds." Mimeographed notes, Princeton University, 1979. Available online (msri.org/publications/books/gt3m/).
  • W. P. Thurston. "Hyperbolic Geometry and 3-Manifolds." In Low-Dimensional Topology" (Bangor, 1979), London Math. Soc. Lecture Note Ser. 48, pp. 9–25. Cambridge, UK: Cambridge University Press, 1982.
  • W. P. Thurston. Three-Dimensional Geometry and Topology, vol. 1. Princeton: Princeton University Press, 1997.
  • J. L. Tollefson. "Involutions of Sufficiently Large 3-Manifolds." Topology 20 (1981), 323–352.
  • A. Ushijima. "The Tilt Formula for Generalized Simplices in Hyperbolic Space." Discrete Comput. Geom. 28 (2002), 19–27.
  • J. R. Weeks. "Convex Hulls and Isometries of Cusped Hyperbolic 3-Manifolds." Topology Appl. 52 (1993), 127–149.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Order Reprints Request Corporate Permissions

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

Request Academic Permissions

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.

Related research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.