Algorithms for recognizing knots and 3-manifolds

Chaos, Solitons and Fractals

Volumn 9, Issue 4-5, 1998, Pages 569-581

  Hass, Joel ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; COMPUTATIONAL GEOMETRY; COMPUTATIONAL METHODS; GRAPH THEORY; THEOREM PROVING; TURING MACHINES;

KNOT THEORY; THREE DIMENSIONAL MANIFOLDS; TOPOLOGICAL ALGORITHMS;

CHAOS THEORY;

EID: 0032048658     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0960-0779(97)00109-4     Document Type: Article

References (29)

1. 1. Hopcroft, J. and Ullman, J., Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading, MA 1979.
2. 2. Rotman, J. J., The Theory of Groups, 2nd edn. Allyn & Bacon, Boston, MA, 1973.
3. 3. Markov, A. A., Insolubility of a problem of homeomorphy. In Proceedings of an International Congress of Mathematicians. Cambridge University Press, 1958, pp. 300-306.
4. 4. Moise, E. E., Affine Structures in 3-manifolds V. Ann. Math., 1952, 55, 96-114.
5. 5. Meeks, W. H. and Yau, S. T., Topology of three dimensional manifolds and the embedding theorems in minimal surface theory. Ann. Math., 1980, 112, 441-484.
6. 6. Meeks, W., Simon, L. and Yau, S. T., Embedded minimal surfaces, exotic spheres and manifolds of positive Ricci curvature. Ann. Math., 1982, 116, 621-659.
7. 7. Thurston, W., The geometry and topology of 3-dimensional manifolds. Princeton University Lecture Notes, 1978.
8. 8. Kneser, H. Geschlossene Flachen in dreidimesionalen Mannifgfaltigkeiten. Jber. Ent. Math., Ver., 1929, 28, 248-260.
9. 9. Papakyriakopoulous, C. D., On Dehn's lemma and the asphericity of knots. Ann. Math., 1957, 66, 1-26.
10. 10. Hempel, J., 3-manifolds. Ann. Math. Stud., 1976, 86.
11. 11. Haken, W., Connections between topological and group theoretical decision problems. In Word Problems, ed. W. W. Boone. North Holland, Amsterdam, 1973, pp. 427-441.
12. 12. Haken, W., Theorie der Normalflachen. Acta Math., 1961, 105, 245-375.
13. 13. Schubert, H., Bestimmung der Primfaktorzerlegung von Verkettungen. Math. Z., 1961, 76,116-148.
14. 14. Jaco, W. and Oertel, U., An algorithm to decide if a 3-manifold is a Haken manifold. Topology, 1984, 23, 2 195-209.
15. 15. Jaco, W. and Tollefson, J. L., Algorithms for the complete decomposition of a closed 3-manifold. Illinois J. Math., 1995, 39, 358-406.
16. 16. Jaco, W. and Rubinstein, J. H., A piecewise linear theory of minimal surfaces in 3-manifolds. J. Diff. Geom., 1988, 27, 493-524.
17. 17. Hempel, J., Residual finiteness for 3-manifolds, Combinatorial group theory and topology. Ann. Math. Stud., 1987. 111, 379-396.
18. 18. Hemion, G., The Classification of Knots and 3-dimensional Spaces. Oxford University Press, 1992.
19. 19. Rubinstein, J. H., Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3-dimensional manifolds. In Proceedings of the Georgia Topology Conference, Am. Math. Soc./Intl Press, 1993, pp. 1-20.
20. 20. Rubinstein, J. H., An algorithm to recognize the 3-sphere. To appear in Proceedings of the ICM, Zurich, 1994, p. 3.
21. 3. Math. Res. Lett., 1994, 1, 613-630.
22. 22. Stocking, M., Almost normal surfaces in 3-manifolds. Ph.D. Thesis, UC Davis, 1996.
23. 23. Sela, Z., The isomorphism problem for hyperbolic groups I. Ann. Math., 1995, 141, 217-283.
24. 24. Johannson, K., Classification problems in low-dimensional topology. In Geometric and Algebraic Topology, Banach Center Publ. 18. PWN, Warsaw, 1986, pp. 37-59.
25. 25. Matveev, S. V., Algorithms for the recognition of the three-dimensional sphere (after A. Thompson). Mat. Sb., 1995, 186, 69-84.
26. 26. Waldhausen, F., Recent results on sufficiently large 3-manifolds. In Proceedings of the Symposium in Pure Mathematics, Vol. 32. Am. Math. Soc., 1978, pp. 21-38.
27. 27. Waldhausen, F., Some problems on 3-manifolds. In Proceedings of the Symposium in Pure Mathematics, Vol. 32. Am. Math. Soc., 1978, pp. 313-322.
28. 28. Birman, J. and Hirsch, M. D., Recognizing the unknot. Preprint 1997.
29. 29. Hass, J., Lagarias, J. C. and Pippenger, N., The computational complexity of knot and link problems. Proc. 38th Annual Symposium On Foundations of Computer Science, 1997, pp. 172-181.