The shapes of physical trefoil knots

Johanns, Paul; Grandgeorge, Paul; Baek, Changyeob; Sano, Tomohiko G.; Maddocks, John H.; Reis, Pedro

doi:10.1016/j.eml.2021.101172

Cited by 20 publications

(14 citation statements)

References 26 publications

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“…The trefoil knot in ref. [51] has the following Kauffman bracket skein relationships, given here by:…”

Section: Spinning Topological Solitons (Cosmological Strings)mentioning

confidence: 99%

A Short Survey of Time Machines

Eastmond¹

2022

Preprint

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In this paper a short survey of time machines (TM) is presented. Mathematical models such as Gödel’s spacetime, Kerr Spacetime, e.t.c., are discussed with emphasis on back in time (BIT) travel. In consummation, we give a representation of a modified source function, based on: (i ) Jauregui-Tsallis’ Conjecture and (ii) the fractional Dirac distribution contructed via Mandlebrot scaling. Notwithstanding that each representation has disparate dimensionality; i.e., d is not qual to D. Thus under cetain limiting values; q-parameter confinement and/or variance constraint, each TM is universally switched on for a given d/D. This means the original energy-momentum tensor (EMT) is recovered.

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“…The trefoil knot in ref. [51] has the following Kauffman bracket skein relationships, given here by:…”

Section: Spinning Topological Solitons (Cosmological Strings)mentioning

confidence: 99%

A Short Survey of Time Machines

Eastmond¹

2022

Preprint

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“…The topology of knots has long been a topic of mathematical interest because it uniquely incorporates geometry and noncommutative algebra (47), and researchers have discovered, for example, that even in two similarly configured knots, a slightly different twist can lead to diametrically opposite stabilities (48)(49)(50). Mechanics-based studies on physical tight knots have revealed the importance of accounting for constituent material properties in knot failure predictions (51)(52)(53), and analyses of loose knots show their potential to increase energy dissipation and introduce stable tightening and untying mechanisms through careful selection of knot geometry and constituent materials (54)(55)(56)(57).…”

Section: Introductionmentioning

confidence: 99%

Knots are not for naught: Design, properties, and topology of hierarchical intertwined microarchitected materials

et al. 2023

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Lightweight and tough engineered materials are often designed with three-dimensional hierarchy and interconnected structural members whose junctions are detrimental to their performance because they serve as stress concentrations for damage accumulation and lower mechanical resilience. We introduce a previously unexplored class of architected materials, whose components are interwoven and contain no junctions, and incorporate micro-knots as building blocks within these hierarchical networks. Tensile experiments, which show close quantitative agreements with an analytical model for overhand knots, reveal that knot topology allows a new regime of deformation capable of shape retention, leading to a ~92% increase in absorbed energy and an up to ~107% increase in failure strain compared to woven structures, along with an up to ~11% increase in specific energy density compared to topologically similar monolithic lattices. Our exploration unlocks knotting and frictional contact to create highly extensible low-density materials with tunable shape reconfiguration and energy absorption capabilities.

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“…They are found everywhere at different length scales from sub-oceanic cables to human-sized rods and ropes, to plant and animal microstructural fabrics, to molecular pillars and chains such as carbon nanotubes and double-stranded deoxyribonucleic acids (DNAs). Their unique large deformability have received attention from the scientific community including the field of theoretical and applied mechanics, and even at present the problems associated with the intricate deformation into three-dimensional (3D) configurations are being tackled in areas such as the compressive buckling from 2D into 3D architecture 1 – 3 , the coiling of ealstic filaments deployed on substrate 4 , 5 , the mechanics of knots 6 , 7 , and the 3D growing rods 8 – 12 .…”

Section: Introductionmentioning

confidence: 99%

“…They are found everywhere at different length scales from suboceanic cables to human-sized rods and ropes, to plant and animal microstructural fabrics, to molecular pillars and chains such as carbon nanotubes and double-stranded deoxyribonucleic acids (DNAs). Their unique large deformability have received attention from the scientific community including the field of theoretical and applied mechanics, and even at present the problems associated with the intricate deformation into three-dimensional (3D) configurations are being tackled in areas such as the compressive buckling from 2D into 3D architecture 1-3 , the coiling of ealstic filaments deployed on substrate 4,5 , the mechanics of knots 6,7 , and the 3D growing rods [8][9][10][11][12] .The deformation of a slender elastic body is concisely described by the elastic rod model represented by a single arc-length parameter prescribing the center line 13,14 . Depending on the elastic rod, there are four types of deformations, i.e., stretching/compression, shearing, bending, and twisting, which are paired with the axial and shear forces and the bending and torsion moments induced inside the rod.…”

mentioning

confidence: 99%

Conformational deformation of a multi-jointed elastic loop

Tanaka

Seki

Ueno

et al. 2022

Sci Rep

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A new class of deformation is presented for a planar loop structure made up of slender elastic bodies and joints. In demonstrating the circumferential shortening of the multi-jointed elastic loop, diverse three-dimensional (3D) deformations emerge through piecewise deflections and discrete rotations. These 3D morphologies correspond to conformations of molecular ring systems. Through image processing, the 3D reconstructions of the deformed structures are characterized by number, geometry, and initial imperfections of the body segments. We elucidate from measurements that the conformational deformation without self-stress results from a cyclical assembly of compressive bending of elastic bodies with high shear rigidity. The mechanical insights gained may apply in controlling the polymorphism exhibited by the cyclical structures across scales.

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The shapes of physical trefoil knots

Cited by 20 publications

References 26 publications

A Short Survey of Time Machines

A Short Survey of Time Machines

Knots are not for naught: Design, properties, and topology of hierarchical intertwined microarchitected materials

Conformational deformation of a multi-jointed elastic loop

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