Recognizing knots using simulated annealing
Abstract
This paper describes the development and implementation of an algorithm which uses simulated annealing to recognize knots by minimizing an energy function defined over all knots. A knot is represented by a piecewise linear curve and the vertices of this curve are perturbed using simulated annealing to minimize the energy. Moving one line segment through another line segment is prohibited. The resulting minimum energy configuration is defined to be the canonical form. The algorithm is then tested with two different types of energy over a collection of complex knots.
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Early knot untying approaches in topological simulation include random perturbation, annealing schedule, energy minimisation with gradient descent, and motion planning.
“The early KU approaches in topological simulation include random perturbation [276,277], annealing schedule [278], energy minimisation with gradient descent [279] and motion planning [280,281].”
From Data-Driven Robotic Manipulation of Cloth-like Deformable Objects: The Present, Challenges and Future Prospects (2023) · cites this paper as [278]
Cited by
Sorted by citations
Nonlinear Dynamics of Stiff Polymers
Goldstein, R. E., & Langer, S. A. (1995) · Physical Review Letters · doi:10.1103/physrevlett.75.1094 · Cited by 146 · see citing works
Interactive topological drawing
Scharein, R. G. (2009) · Open Collections · doi:10.14288/1.0051670 · Cited by 77 · see citing works
Using Motion Planning for Knot Untangling
Ladd, A. M., & Kavraki, L. E. (2004) · The International Journal of Robotics Research · doi:10.1177/0278364904045469 · Cited by 56 · see citing works
“Other examples of energy functions can be found in Ligocki and Sethian (1994), Diao, Ernst and Van Rensburg (1997), Grzeszczuk, Huang and Kauffman (1997), and Scharein (1988), and include geometric energy, Möbius energy, spring energy, electrostatic energy and others.”
BackgroundMethodInfluentialMÖBIUS-INVARIANT KNOT ENERGIES
Kusner, R., & Sullivan, J. M. (1998) · Series on knots and everything · doi:10.1142/9789812796073_0017 · Cited by 34 · see citing works
Kadi, H. A., & Terzić, K. (2023) · Sensors · doi:10.3390/s23052389 · Cited by 19 · see citing works
“The early KU approaches in topological simulation include random perturbation [276,277], annealing schedule [278], energy minimisation”
MethodMotion Planning for Knot Untangling
Ladd, A. M., & Kavraki, L. E. (2004) · Springer tracts in advanced robotics · doi:10.1007/978-3-540-45058-0_2 · Cited by 15 · see citing works
Untangling knots by stochastic energy optimization
Huang, M., Grzeszczuk, R. P., & Kauffman, L. H. (1996) · IEEE Visualization · doi:10.5555/244979.245605 · Cited by 14 · see citing works
Physically-based stochastic simplification of mathematical knots
Grzeszczuk, R., Huang, M., & Kauffman, L. H. (1997) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/2945.620492 · Cited by 11 · see citing works
On a flow to untangle elastic knots
Lin, C.-C., & Schwetlick, H. (2010) · Calculus of Variations and Partial Differential Equations · doi:10.1007/s00526-010-0328-0 · Cited by 8 · see citing works
Operator Formalism for Topology-Conserving Crossing Dynamics in Planar Knot Diagrams
Rohwer, C. M., & Müller-Nedebock, K. K. (2015) · Journal of Statistical Physics · doi:10.1007/s10955-014-1161-1 · Cited by 8 · see citing works
“) Various uphill perturbation methods have been used, including placing point charges near the (charged) knot and allowing it to evolve under electrostatic forces, tightening or loosening various parts of the knot [17], and perturbing the vertices of a piecewise linear curve representation of the knot [60].”
BackgroundMethodAnalysis of the first variation and a numerical gradient flow for integral Menger curvaturePreprint
Hermes, T. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1408.6517 · Cited by 7 · see citing works
Operator Formalism for Topology-Conserving Crossing Dynamics in Planar Knot Diagrams
Rohwer, C. M., & Müller-Nedebock, K. K. (2015) · MPG.PuRe (Max Planck Society) · Cited by 5 · see citing works
Kadi, H. A., & Terzić, K. (2022) · Preprints.org · doi:10.20944/preprints202212.0305.v1 · Cited by 1 · see citing works
Combinatorics and dynamics in polymer knots
Rohwer, C. M. (2014) · SUNScholar (Stellenbosch University)
Untangling Mathematical Knots with Simulated Annealing and Opposition-Based Learning
Lin, J., & Zhang, H. (2018) · IEEE International Conference on Big Data (BigData) · doi:10.1109/bigdata.2018.8622560
“The third one is to transform one knot to its isotopic structure through a sequence of conformation-maintaining movements, which leads to the development and use of many approximate models and integration methods [6] [10] [11] [12] [13].”
Background
- Authors
- Ligocki, T. J., & Sethian, J. A.
- Affiliations
- Lawrence Berkeley National Laboratory (US)
- ORCID
- Sethian, James A.
- Venue
- Journal of Knot Theory and Its Ramifications
- Publisher
- World Scientific Pub Co Pte Lt
- Published
- 1994-12
- Volume
- 03
- Issue
- 04
- Pages
- 477-495
- ISSN
- 0218-2165 (ISSN-L)
- 1793-6527
- OpenAlex
- W2059132453
- Cited by
- 15
- References
- 0
- Citations by year
- 2023: 1 · 2022: 1 · 2018: 1 · 2015: 2 · 2014: 2
- FWCI
- 1.75
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Knot Theory and Its Ramifications (publishedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- eScholarship (California Digital Library) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Computer Science › Computer Graphics and Computer-Aided Design › Computer Graphics and Visualization Techniques
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Digital Image Processing Techniques
- Keywords
- simulated annealing, energy minimization, piecewise linear curves, canonical form
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