To conclude, the above analysis shows how
basic topological counting rules can be used
to estimate the relative stability of frequently
encountered knots and tangles. From a theo-
retical perspective, the parallels with long-
range coupled spin systems suggest that the
statistical mechanics ( 4 , 21 )ofgeneralknotted
structurescanbeformulatedwithinthisframe-
work. Tangled vortices ( 12 , 17 ) in complex fluids
and defect loops in liquid crystals ( 15 ) may
permit similar statistical descriptions through
reduction to topological crossing diagrams. In
elastic systems, joint experimental and theo-
retical progress is needed to untangle long-
standing puzzles regarding the statistics of
knots in DNA ( 18 ) and proteins ( 20 , 21 ), where
thermal effects induce a finite persistence
length, and other macroscopic structures
( 8 , 24 ). In sailing, climbing, and many other
applications, nontopological material param-
eters and contact geometry ( 30 )alsoplayim-
portant mechanical roles and must be included
in more refined continuum models to quanti-
tatively describe practically relevant knotting
phenomena. From a broader conceptual and
practical perspective, the above topological
mechanics framework seems well suited for
designing and exploring new classes of knots
with desired behaviors under applied load.
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ACKNOWLEDGMENTS
J.D. thanks the Isaac Newton Institute for Mathematical Sciences for
support and hospitality during the program“The Mathematical Design
of New Materials”(supported by EPSRC grant EP/R014604/1) when
work on this paper was undertaken. We thank J. Takagi for producing
Fig. 4A.Funding:This work was supported by an Alfred P. Sloan
Research Fellowship (J.D.), a Complex Systems Scholar Award from
the James S. McDonnell Foundation (J.D.), the Brigham and
Women’s Hospital through a Stepping Strong Innovator Award
(J.D.S. and M.K.), and the National Science Foundation through the
“Designing Materials to Revolutionize and Engineer our Future”
program (DMREF-1922321 to J.D.S. and M.K.).Author contributions:
V.P.P. and J.D. developed theory. V.P.P. performed simulations,
for which J.D.S. and M.K. provided data and code for converting strain
into perceived knot color. J.D.S. and M.K. designed color-changing
fibers and conceived optomechanical experiments. J.D.S. conducted
optomechanical experiments and provided the description of the
experiments in the supplementary materials. J.D.S. and V.P.P.
performed in-lab prisoner’s escape experiments. V.P.P. and J.D. wrote
the first draft of the paper. All authors discussed and revised the
manuscript.Competing interests:The authors declare no competing
interests.Data and materials availability:The code used for
numerical simulations is available on Zenodo ( 31 ). All data are available
in the main text or the supplementary materials.SUPPLEMENTARY MATERIALS
science.sciencemag.org/content/367/6473/71/suppl/DC1
Materials and Methods
Supplementary Text
Figs. S1 to S6
Tables S1 and S2
Movies S1 and S2
References ( 32 – 33 )
5 August 2019; accepted 15 November 2019
10.1126/science.aaz0135Patilet al.,Science 367 ,71–75 (2020) 3 January 2020 5of5
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