Knot tightening by constrained gradient descent
Open accessAbstract
We present new computations of approximately length-minimizing polygons with fixed thickness. These curves model the centerlines of “tight” knotted tubes with minimal length and fixed circular cross-section. Our curves approximately minimize the ropelength (or quotient of length and thickness) for polygons in their knot types. While previous authors have minimized ropelength for polygons using simulated annealing, the new idea in our code is to minimize length over the set of polygons of thickness at least one using a version of constrained gradient descent. We rewrite the problem in terms of minimizing the length of the polygon subject to an infinite family of differentiable constraint functions. We prove that the set of variations of a polygon of thickness one that does not decrease thickness to first order is a finitely generated polyhedral cone, and give an explicit set of generators. Using this cone, we give a first-order minimization procedure and a Karush–Kuhn–Tucker criterion for polygonal-ropelength criticality. Our main numerical contribution is a set of 379 almost-critical knots and links, including all prime knots with ten and fewer crossings and all prime links with nine and fewer crossings. For links, these are the first published ropelength figures, and for knots they improve on existing figures. We give new maps of the self-contacts of these knots and links, and discover some highly symmetric tight knots with particularly simple-looking self-contact maps.
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Many complex knots used for connecting and decorations must be tightened at specific locations along the rope by leveraging friction locks.
“However, many complex knots used for connecting and decorations must be tightened at specific locations along the rope [288] by leveraging friction locks [289,290,291,292], i.e., locations where friction resits the rope against external forces.”
From Data-Driven Robotic Manipulation of Cloth-like Deformable Objects: The Present, Challenges and Future Prospects (2023) · cites this paper as [291]
The Introduction (page unavailable) describes knots tied in rope as binding tightly and introduces a computational method for knot-tightening. It does not say that connecting or decorative knots must be tightened at specific locations along a rope. The friction-lock explanation is supported by separate citations in the citing sentence, not by this source. (LLM, conf=0.82)
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Research works printed but not linked: 1
- Eric Rawdon. The Thickness of Polygonal Knots. PhD thesis, The University of Iowa, 1997.
Cited by
Sorted by citations
Kadi, H. A., & Terzić, K. (2023) · Sensors · doi:10.3390/s23052389 · Cited by 19 · see citing works
- Authors
- Ashton, T., Cantarella, J., Piatek, M., & Rawdon, E. J.
- Affiliations
- University of Georgia (US)
- University of Washington (US)
- University of St. Thomas - Minnesota (US)
- Venue
- Experimental Mathematics
- Publisher
- Informa UK Limited
- Published
- 2011-03-28
- Volume
- 20
- Issue
- 1
- Pages
- 57-90
- ISSN
- 1058-6458 (ISSN-L)
- 1944-950X
- OpenAlex
- W2108273387
- Cited by
- 71
- References
- 53
- Citations by year
- 2026: 2 · 2025: 2 · 2024: 8 · 2023: 5 · 2022: 3 · 2021: 6 · 2020: 2 · 2019: 3 · 2018: 3 · 2017: 3 · 2016: 3 · 2015: 7 · 2014: 11 · 2013: 5 · 2012: 5
- FWCI
- 9.29
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Experimental Mathematics (publishedVersion)
- Project Euclid (Cornell University) (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- UST Research Online (University of St. Thomas - Minnesota) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation
- 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
- ropelength, Karush-Kuhn-Tucker conditions, prime knots, prime links
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