Using Motion Planning for Knot Untangling

In this paper we investigate the application of motion planning techniques to the untangling of mathematical knots. Knot untangling can be viewed as a high-dimensional planning problem in reparametrizable configuration spaces. In the past, simulated annealing and other energy minimization methods have been used to find knot untangling paths. We have developed a probabilistic planner that is capable of untangling knots described by over 400 variables. We have tested on known difficult benchmarks in this area and untangled them more quickly than has been achieved with minimization in the literature. In this work, the use of motion planning techniques is critical for the untangling. Our planner defines local goals and makes combined use of energy minimization and randomized tree-based planning. We also show how to produce candidates with a minimal number of segments for a given knot. The planner developed in this work is novel in that it is used to study issues arising in practical motion planning for high-dimensional and reparametrizable geometry. The use of energy methods, local goals and tree-based expansion is also novel and may suggest solutions in other planning applications. Finally, we discuss some possible applications of our untangling planner in computational topology, in the study of DNA rings and protein folding and for planning with flexible robots.

[1]  S. Whittington,et al.  Knots in self-avoiding walks , 1988 .

[2]  Nicholas Pippenger,et al.  Knots in random walks , 1989, Discret. Appl. Math..

[3]  D. Welsh,et al.  On the computational complexity of the Jones and Tutte polynomials , 1990, Mathematical Proceedings of the Cambridge Philosophical Society.

[4]  J. Wang,et al.  Knotting of a DNA chain during ring closure. , 1993, Science.

[5]  N. Cozzarelli,et al.  Processive recombination by wild-type gin and an enhancer-independent mutant. Insight into the mechanisms of recombination selectivity and strand exchange. , 1994, Journal of molecular biology.

[6]  L. Kauffman,et al.  GRAPH INVARIANTS AND THE TOPOLOGY OF RNA FOLDING , 1994 .

[7]  S. Y. Shaw,et al.  DNA KNOT FORMATION IN AQUEOUS SOLUTIONS , 1994 .

[8]  J. Simon.  ENERGY FUNCTIONS FOR POLYGONAL KNOTS , 1994 .

[9]  J. Sethian,et al.  Recognizing Knots Using Simulated Annealing , 1994 .

[10]  J. Dubochet,et al.  Electrophoretic mobility of DNA knots , 1996, Nature.

[11]  Louis H. Kauffman,et al.  Untangling knots by stochastic energy optimization , 1996, Proceedings of Seventh Annual IEEE Visualization '96.

[12]  B. Faverjon,et al.  Probabilistic Roadmaps for Path Planning in High-Dimensional Con(cid:12)guration Spaces , 1996 .

[13]  J. Dubochet,et al.  Geometry and physics of knots , 1996, Nature.

[14]  Louis H. Kauffman,et al.  Physically-Based Stochastic Simplification of Mathematical Knots , 1997, IEEE Trans. Vis. Comput. Graph..

[15]  W. Lickorish,et al.  An Introduction to Knot Theory , 1997 .

[16]  Y. Diao,et al.  In Search of a Good Polygonal Knot Energy , 1997 .

[17]  J. Lagarias,et al.  The number of Reidemeister moves needed for unknotting , 1998, math/9807012.

[18]  J. Weeks,et al.  The first 1,701,936 knots , 1998 .

[19]  J. Little,et al.  Interactive topological drawing , 1998 .

[20]  B. Peter,et al.  Knot what we thought before: the twisted story of replication. , 1999, BioEssays : news and reviews in molecular, cellular and developmental biology.

[21]  Leonidas J. Guibas,et al.  Emerging Challenges in Computational Topology , 1999, ArXiv.

[22]  Jeffrey C. Lagarias,et al.  The computational complexity of knot and link problems , 1999, JACM.

[23]  C. Aneziris.  The mystery of knots : computer programming for knot tabulation , 1999 .

[24]  Rajeev Motwani,et al.  Path Planning in Expansive Configuration Spaces , 1999, Int. J. Comput. Geom. Appl..

[25]  Steven M. LaValle,et al.  Rapidly-Exploring Random Trees: Progress and Prospects , 2000 .

[26]  X. Dai,et al.  THE MINIMUM OF KNOT ENERGY FUNCTIONS , 2000 .

[27]  Steven M. LaValle,et al.  RRT-connect: An efficient approach to single-query path planning , 2000, Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065).

[28]  C. Benham,et al.  WRITHING AND LINKING DENSITIES FOR CLOSED, CIRCULAR DNA , 2000 .

[29]  David Hsu,et al.  Randomized single-query motion planning in expansive spaces , 2000 .

[30]  I. Darcy.  BIOLOGICAL DISTANCES ON DNA KNOTS AND LINKS: Applications to XER recombination , 2001 .

[31]  P. Boldi,et al.  TOWARDS AN IMPLEMENTATION OF THE B–H ALGORITHM FOR RECOGNIZING THE UNKNOT , 2001, math/0102177.

[32]  E. L. Zechiedrich,et al.  Topoisomerase IV, alone, unknots DNA in E. coli. , 2001, Genes & development.

[33]  Jean-Claude Latombe,et al.  Capturing molecular energy landscapes with probabilistic conformational roadmaps , 2001, Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No.01CH37164).

[34]  Lydia E. Kavraki,et al.  Planning Paths for Elastic Objects under Manipulation Constraints , 2001, Int. J. Robotics Res..

[35]  Bruce Randall Donald,et al.  Algorithmic and Computational Robotics: New Directions , 2001 .

[36]  S. LaValle,et al.  Randomized Kinodynamic Planning , 2001 .

[37]  Lydia E. Kavraki,et al.  A dimensionality reduction approach to modeling protein flexibility , 2002, RECOMB '02.

[38]  Jean-Claude Latombe,et al.  Stochastic roadmap simulation: an efficient representation and algorithm for analyzing molecular motion , 2002, RECOMB '02.

[39]  Nancy M. Amato,et al.  Using Motion Planning to Study Protein Folding Pathways , 2002, J. Comput. Biol..

[40]  Nancy M. Amato,et al.  Choosing good paths for fast distributed reconfiguration of hexagonal metamorphic robots , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[41]  Lydia E. Kavraki,et al.  Simulated knot tying , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[42]  Nancy M. Amato,et al.  Probabilistic roadmap motion planning for deformable objects , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[43]  Joel Hass,et al.  3-MANIFOLD KNOT GENUS is NP-complete , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[44]  Sergei Vassilvitskii,et al.  A complete, local and parallel reconfiguration algorithm for cube style modular robots , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[45]  Jean-Claude Latombe,et al.  Stochastic Roadmap Simulation: An Efficient Representation and Algorithm for Analyzing Molecular Motion , 2003, J. Comput. Biol..

[46]  Andrew M. Ladd,et al.  Measure Theoretic Analysis of , 2004 .