Physically-based stochastic simplification of mathematical knots

IEEE Transactions on Visualization and Computer Graphics

Volumn 3, Issue 3, 1997, Pages 262-272

  Grzeszczuk, Robert P ^a   Huang, Milana ^a   Kauffman, Louis H ^a  

^a HEWLETT PACKARD ENTERPRISE   (United States)

Author keywords

Knot classification; Knot theory; Mathematical visualization; Polynomial invariants; Simulated annealing

Indexed keywords

FUNCTIONS; MATHEMATICAL MODELS; POLYNOMIALS; RANDOM PROCESSES; SIMULATED ANNEALING; VISUALIZATION;

ENERGY FUNCTIONS; MATHEMATICAL KNOTS;

COMPUTATIONAL GEOMETRY;

EID: 0031186232     PISSN: 10772626     EISSN: None     Source Type: Journal    
DOI: 10.1109/2945.620492     Document Type: Article

References (25)

1. J.W. Alexander, "Topological Invariants of Knots and Links," Trans. Am. Math. Soc., vol. 20, pp. 275-306,1923.
2. P. Bevington, Data Reduction and Error Analysis for the Physical Sciences. McGraw-Hill, 1992.
3. S. Bryson, M. Freedman, Z.-X. He, and Z. Wang, "Mobius Invariance of Knot Energy," Bull. Am. Math. Soc., vol. 28, pp. 99-103, 1993.
4. G. Buck and J. Simon, "Knots as Dynamical Systems," Topology Appl, vol. 51,1993.
5. G.G. Fox, R.D. Williams, and P.C. Messina, Parallel Computing Works. Morgan Kaufmann Publishers, Inc., 1994.
6. M.H. Freedman, Z.-X. He, and Z. Wang, "On the Energy of Knots and Unknots," Annals of Math., vol. 139, pp. 1-50, 1994.
7. S. Fukuhara, "Energy of a Knot," A Fete of Topology, Y. Matsumoto, T. Mizutani, and S. Morita, eds., pp. 443-451. Academic Press, Inc., 1988.
8. C.F. Gauss, Werke. Teubner, Leipzig, 1900.
9. M. Huang, R. Grzeszczuk, and L. Kauffman, "Untangling Knots by Stochastic Energy Optimization," Proc. IEEE Conf. Visualization, pp. 279-286,1996.
10. V. Interrante, H. Fuchs, and S. Pizer, "Illustrating Transparent Surfaces with Curvature-Directed Strokes," Proc. Visualization '96 Conf., pp. 211-218,1996.
11. L. Kauffman, M. Huang, and R.P. Grzeszczuk, "Self-Repelling Knots and Local Energy Minima," Proc. 1996 Conf. Random Knotting, 1996.
12. L.H. Kauffman, Knots and Physics. World Scientific Publishing, 1993.
13. S. Kirkpatrick, "Optimization by Simulated Annealing: Quantitative Studies," J. Statistical Physics, vol. 34, pp. 975-986,1984.
14. S. Kirkpatrick, C.D. Gelatt, and M.P. Vecchi, "Optimization by Simulated Annealing," Science, vol. 220, pp. 671-680,1983.
15. R.B. Kusner and J.M. Sullivan, "Mobius Energies for Knots and Links, Surfaces and Submanifolds," Geometric Topology, W.H. Kazez, éd., pp. 570-604. Am. Math. Soc., International Press, 1997.
16. T. Ligocki and J. Sethian, "Recognizing Knots Using Simulated Annealing," J. Knot Theory and Its Ramifications, pp. 477-495,1994.
17. M. Ochiai, "Non-Trivial Projections of the Trivial Knot," S.M.F. Astérisque, vol. 192, pp. 7-9, 1990.
18. J. O'Hara, "Energy of a Knot," Topology, vol. 30, pp. 241-247, 1991.
19. W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes in C. Cambridge Univ. Press, 1988.
20. R.C. Read and P. Rosenstiehl, "On the Gauss Crossing Problem," Colloquia Math. Soc. }. Bolyai, Combinatorics, vol. 18, pp. 843-876, 1976.
21. K. Reidemeister, Knotentheorie. Chelsea Publishing Co., 1948.
22. P. Rheingans, "Opacity-Modulating Triangular Textures for Irregular Surfaces," Proc. Visualization '96 Conf., pp. 219-225,1996.
23. H. Shu and R. Hartley, "Fast Simulated Annealing," Phys. Lett. A., vol. 122, pp. 157-162,1987.
24. J. Simon, "Energy Functions for Polygonal Knots," Random Knotting and Linking. World Scientific Publishing, 1994.
25. Y.-Q. Wu, personal correspondence, Jan. 1996.