Energy functions for polygonal knots
Abstract
We define a scale-invariant energy function for polygonal knots in ℜ3 based on the minimum distances between segments. The energy is bounded below by 2π. (minimum crossing number of the knot type). For each knot type, there exists an ideal number of segments, from which can be made an ideal conformation of the knot having minimum energy among all polygons realizing that knot type. Results leading to this include the following: The energy of an n-segment polygon is greater than n; if energy is bounded then ratios of edge lengths and angles are bounded away from zero; changing knot type requires passing an infinite energy barrier. We implement an algorithm, with the feature that preserving knot-type is guaranteed, to discover local energy-minimizing conformations for a given number of segments, and present some of these examples.
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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 [276]
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Curvature and torsion of the tight closed trefoil knot
Baranska, J., Przybyl, S., & Pieranski, P. (2008) · The European Physical Journal B · doi:10.1140/epjb/e2008-00443-y · Cited by 20 · 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 with gradient descent [279] and motion planning [280,281].”
Method
- Authors
- SIMON, J. K.
- Affiliations
- University of Iowa (US)
- Corresponding author
- SIMON, JONATHAN K.
- Venue
- Journal of Knot Theory and Its Ramifications
- Publisher
- World Scientific Pub Co Pte Lt
- Published
- 1994-09
- Volume
- 03
- Issue
- 03
- Pages
- 299-320
- ISSN
- 0218-2165 (ISSN-L)
- 1793-6527
- OpenAlex
- W2137745368
- Cited by
- 59
- References
- 9
- Citations by year
- 2026: 4 · 2024: 2 · 2023: 2 · 2022: 2 · 2021: 1 · 2020: 5 · 2019: 4 · 2018: 1 · 2016: 1 · 2015: 3 · 2014: 1
- FWCI
- 2.91
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Knot Theory and Its Ramifications (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Mathematics › Geometry and Topology › Mathematics and Applications
- Keywords
- polygonal knots, knot energy, energy barrier, edge length ratio
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