The complexity of the normal surface solution space

Proceedings of the Annual Symposium on Computational Geometry

Volumn , Issue , 2010, Pages 201-209

  Burton, Benjamin A ^a  

^a UNIVERSITY OF QUEENSLAND   (Australia)

Author keywords

3 manifolds; Complexity; Computational topology; Normal surfaces; Vertex enumeration

Indexed keywords

3-MANIFOLDS; COMPREHENSIVE ANALYSIS; COMPUTATIONAL BOTTLENECKS; COMPUTATIONAL TOPOLOGY; EXPONENTIAL BOUNDS; SOLUTION SPACE; SURFACE THEORY; THEORY AND PRACTICE; THREE DIMENSIONS; TOPOLOGICAL ALGORITHMS; UPPER BOUND; VERTEX ENUMERATION; VERTEX NORMAL; WORST CASE SCENARIO;

ALGORITHMS; COMPUTATIONAL COMPLEXITY; TOPOLOGY; TRIANGULATION;

COMPUTATIONAL GEOMETRY;

EID: 77954921903     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/1810959.1810995     Document Type: Conference Paper

References (37)

1. I. Agol, J. Hass, and W. Thurston. 3-manifold knot genus is NP-complete. In STOC '02: Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, pages 761-766. ACM Press, 2002. (Pubitemid 35009439)
2. D. Avis, D. Bremner, and R. Seidel. How good are convex hull algorithms? Comput. Geom., 7(5-6):265-301, 1997.
3. B. A. Burton. Regina: Normal surface and 3-manifold topology software. http://regina.sourceforge.net/, 1999-2009.
4. B. A. Burton. Introducing Regina, the 3-manifold topology software. Experiment. Math., 13(3):267-272, 2004.
5. B. A. Burton. Enumeration of non-orientable 3-manifolds using face-pairing graphs and union-find. Discrete Comput. Geom., 38(3):527-571, 2007.
6. B. A. Burton. Structures of small closed non-orientable 3-manifold triangulations. J. Knot Theory Ramifications, 16(5):545-574, 2007.
7. B. A. Burton. Converting between quadrilateral and standard solution sets in normal surface theory. Algebr. Geom. Topol., 9(4):2121-2174, 2009.
8. B. A. Burton. Quadrilateral-octagon coordinates for almost normal surfaces. To appear in Experiment. Math., arXiv:0904.3041, Apr. 2009.
9. B. A. Burton. Optimizing the double description method for normal surface enumeration. Math. Comp., 79(269):453-484, 2010.
10. B. A. Burton, J. H. Rubinstein, and S. Tillmann. The Weber-Seifert dodecahedral space is non-Haken. Preprint, arXiv:0909.4625, Sept. 2009.
11. P. J. Callahan, M. V. Hildebrand, and J. R. Weeks. A census of cusped hyperbolic 3-manifolds. Math. Comp., 68(225):321-332, 1999.
12. M. E. Dyer. The complexity of vertex enumeration methods. Math. Oper. Res., 8(3):381-402, 1983.
13. B. Grünbaum. Convex Polytopes. Number 221 in Graduate Texts in Mathematics. Springer, New York, 2nd edition, 2003.
14. W. Haken. Theorie der Normalflächen. Acta Math., 105:245-375, 1961.
15. W. Haken. Über das Homöomorphieproblem der 3- Mannigfaltigkeiten. I. Math. Z., 80:89-120, 1962.
16. J. Hass, J. C. Lagarias, and N. Pippenger. The computational complexity of knot and link problems. J. Assoc. Comput. Mach., 46(2):185-211, 1999.
17. W. Jaco and U. Oertel. An algorithm to decide if a 3-manifold is a Haken manifold. Topology, 23(2):195-209, 1984.
18. W. Jaco, H. Rubinstein, and S. Tillmann. Minimal triangulations for an infinite family of lens spaces. J. Topol., 2(1):157-180, 2009.
19. W. Jaco and J. H. Rubinstein. 0-efficient triangulations of 3-manifolds. J. Differential Geom., 65(1):61-168, 2003.
20. W. Jaco, J. H. Rubinstein, and S. Tillmann. Coverings and minimal triangulations of 3-manifolds. To appear in Algebr. Geom. Topol., arXiv:0903.0112, Feb. 2009.
21. W. Jaco and J. L. Tollefson. Algorithms for the complete decomposition of a closed 3-manifold. Illinois J. Math., 39(3):358-406, 1995.
22. L. Khachiyan, E. Boros, K. Borys, K. Elbassioni, and V. Gurvich. Generating all vertices of a polyhedron is hard. Discrete Comput. Geom., 39(1-3):174-190, 2008.
23. H. Kneser. Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten. Jahresbericht der Deut. Math. Verein., 38:248-260, 1929.
24. T. Li. An algorithm to determine the Heegaard genus of a 3-manifold. Preprint, arXiv:1002.1958, Feb. 2010.
25. A. A. Markov. Insolubility of the problem of homeomorphy. In Proc. Internat. Congress Math. 1958, pages 300-306. Cambridge Univ. Press, New York, 1960.
26. B. Martelli and C. Petronio. Three-manifolds having complexity at most 9. Experiment. Math., 10(2):207-236, 2001.
27. W. S. Massey. A Basic Course in Algebraic Topology. Number 127 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1991.
28. S. Matsumoto and R. Rannard. The regular projective solution space of the figure-eight knot complement. Experiment. Math., 9(2):221-234, 2000.
29. S. V. Matveev. Tables of 3-manifolds up to complexity 6. Max-Planck-Institut für Mathematik Preprint Series, (67), 1998. Available from http://www.mpim-bonn.mpg.de/html/preprints/preprints.html.
30. S. V. Matveev. Recognition and tabulation of three-dimensional manifolds. Dokl. Akad. Nauk, 400(1):26-28, 2005.
31. P. McMullen. The maximum numbers of faces of a convex polytope. Mathematika, 17:179-184, 1970.
32. J. H. Rubinstein. An algorithm to recognize the 3-sphere. In Proceedings of the International Congress of Mathematicians (Zürich, 1994), volume 1, pages 601-611. Birkhäuser, 1995.
33. J. H. Rubinstein. Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3-dimensional manifolds. In Geometric Topology (Athens, GA, 1993), volume 2 of AMS/IP Stud. Adv. Math., pages 1-20. Amer. Math. Soc., 1997.
34. 3. Math. Res. Lett., 1(5):613-630, 1994.
35. W. P. Thurston. The geometry and topology of 3-manifolds. Lecture notes, Princeton University, 1978.
36. W. P. Thurston. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.), 6(3):357-381, 1982.
37. J. L. Tollefson. Normal surface Q-theory. Paci-c J. Math., 183(2):359-374, 1998.