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Physically-Based Stochastic Simplification of Mathematical Knots
July-September 1997 (vol. 3 no. 3)
pp. 262-272

Abstract—The article describes a tool for simplification and analysis of tangled configurations of mathematical knots. The proposed method addresses optimization issues common in energy-based approaches to knot classification. In this class of methods, an initially tangled elastic rope is "charged" with an electrostatic-like field which causes it to self-repel, prompting it to evolve into a mechanically stable configuration. This configuration is believed to be characteristic for its knot type. We propose a physically-based model to implicitly guard against isotopy violation during such evolution and suggest that a robust stochastic optimization procedure, simulated annealing, be used for the purpose of identifying the globally optimal solution. Because neither of these techniques depends on the properties of the energy function being optimized, our method is of general applicability, even though we applied it to a specific potential here. The method has successfully analyzed several complex tangles and is applicable to simplifying a large class of knots and links. Our work also shows that energy-based techniques will not necessarily terminate in a unique configuration, thus we empirically refute a prior conjecture that one of the commonly used energy functions (Simon's) is unimodal. Based on these results we also compare techniques that rely on geometric energy optimization to conventional algebraic methods with regards to their classification power.

[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.C. 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, ed., 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. Asterisque, 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. J. 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," Visualization 96, IEEE Computer Society Press, Los Alamitos, Calif., 1996, pp. 219-225.
[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.

Index Terms:
Mathematical visualization, knot theory, knot classification, polynomial invariants, simulated annealing.
Robert P. Grzeszczuk, Milana Huang, Louis H. Kauffman, "Physically-Based Stochastic Simplification of Mathematical Knots," IEEE Transactions on Visualization and Computer Graphics, vol. 3, no. 3, pp. 262-272, July-Sept. 1997, doi:10.1109/2945.620492
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