Knot theory is an interesting area of mathematics studying the unique properties of topological morphs in knot attribute. It is an investigation into the way to extend the one dimensional string to the wider implications in the higher dimensional space. The attached book provides an excellent reference:
In polymer research knot theory is applied. For example: http://dx.doi.org/10.1021/ma300942a
In polymer research knot theory is applied. For example: http://dx.doi.org/10.1021/ma300942a
Hello Christoph, thanks for sharing this exciting idea surrounding the ring polymers and their relatedness to knots. I am looking at a very common life example of how knots can be observed in macroscopic real life objects and their interactions!
In addition to the excellent observation by @Christoph Jentzsch, there is the paper on physical knot theory by K.C. Millett to consider:
K.C. Millett, Physical Knot Theory: An Introduction to the Study of the
Influence of Knotting on the Spatial Characteristics of Polymers, WSPC 2011:
http://www.math.ucsb.edu/~millett/Papers/2011MillettPhysicalKnotsTrieste2009Final.pdf
See Section 3, Elements of physical knot theory, starting on page 7, to consider the heart of the matter.
A very detailed exploration of physical knot theory is given in
R.G. Scharein, Interactive Topological Drawing, Ph.D. thesis, The University of British Columbia, 1998:
file:///Users/jfpeters/Downloads/ubc_1998-272389.pdf
See Section 3.5, starting on page 47, for the specifics of physical knot theory. This is an exceptional exploration, which traces the origin of knot theory in physics and the work by Carl Friedrich Guss (1777-1855), his work on knot theory in his study of thermodynamics. 4-dimensional knot theory is covered in Section 3.6, starting on page 55.
In Phys. Rev. Lett. 111, 150404 (2013), Tying knots in Light Fields, the authors show that knots on torus are solutions of Maxwell equations. As the scale is not specified, then one may suspect ball lightnings are manifestations of those solutions. Maybe they are also good models of elementary particles' internal structure? Those solutions have infinite lifetime (in free space!) but oscillate in time, so there is no friction there.
Not exactly about friction or adhesion but certainly common life example of knots on fishing (and other) ropes:
http://iopscience.iop.org/1367-2630/3/1/310/fulltext/
New J. Phys. 3 (2001) 10
PII: S1367-2630(01)20698-3
Localization of breakage points in knotted strings
Piotr Pieranski, Sandor Kasas, Giovanni Dietler, Jacques Dubochet and Andrzej Stasiak
Abstract: It is a common macroscopic observation that knotted ropes or fishing lines under tension easily break at the knot. However, a more precise localization of the breakage point in knotted macroscopic strings is a difficult task. In the present work, the tightening of knots was numerically simulated, a comparison of strength of different knots was experimentally performed and a high velocity camera was used to precisely localize the site where knotted macroscopic strings break. In the case of knotted spaghetti, the breakage occurs at the position with high curvature at the entry to the knot. This localization results from joint contributions of loading, bending and friction forces into the complex process of knot breakage. The present simulations and experiments are in agreement with recent molecular dynamics
simulations of a knotted polymer chain and with experiments performed on actin and DNA filaments. The strength of the knotted string is greatly reduced (down to 50%) by the presence of a knot, therefore reducing the resistance to tension of all materials containing chains of any sort. The present work with macroscopic strings revels some important aspects, which are not accessible by experiments with microscopic chains.
See also:
Eur. Phys. J. E 4, 445-449 (2001)
arXiv:physics/0103039 v1 14 Mar 2001
http://arxiv.org/abs/0905.3805
http://arxiv.org/abs/1402.5760
In addition to the excellent (& classical) sources already mentioned above:
Brown et al:
http://robotics.stanford.edu/~latombe/papers/knotmaking/paper.pdf
Durville (2012) Contact friction modeling within elastic beam assemblies: an application to knot tightening. DOI 10.1007/s00466-012-0683-0
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