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Published on: May 19, 2014
Spontaneous knotting of an agitated string
Dorian M Raymer1, Douglas E Smith
1Department of Physics, University of California at San Diego, 9500 Gilman Drive, Mail Code 0379, La Jolla, CA 92093, USA. draymer@physics.ucsd.edu
Summary
Complex knots form rapidly in jostled strings. Knot probability increases with length, then saturates, differing from random walks due to finite agitation and stiffness, but approaches 100% for long, flexible strings.
Area of Science:
- Physics
- Mathematics
- Materials Science
Background:
- Spontaneous knot formation in strings is a known phenomenon but poorly understood.
- Factors influencing knot complexity and probability remain unclear.
Purpose of the Study:
- To investigate the spontaneous knotting of strings during agitation.
- To analyze the types and complexity of knots formed using mathematical knot theory.
Main Methods:
- Experimental tumbling of strings in a box.
- Analysis of knot topology using mathematical knot theory, including Jones polynomials.
- Computer analysis of digital images of formed knots.
Main Results:
- Complex knots form within seconds.
- Knotting probability saturates below 100% with increasing string length, unlike random walks.
- 120 distinct prime knot types, up to 11 crossings, were observed.
- Knot formation probability decreases exponentially with knot complexity (crossing number, Möbius energy).
Conclusions:
- Finite agitation time and string stiffness limit knot formation probability.
- Long, flexible strings approach 100% knotting probability.
- A proposed model based on random braid moves qualitatively explains observed knot distributions and dependencies.
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