Minimal unlinking pathways as geodesics in knot polynomial space

Liu, Xin; Ricca, Renzo L.; Li, Xin-Fei

doi:10.1038/s42005-020-00398-y

Cited by 6 publications

(4 citation statements)

References 26 publications

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“…This subsection presents a systematic study on the effect of the torus aspect ratio on the stability (Ricca et al 1999) of coiled vortex loops T 1,q , extending the investigation in § 4.1. Note that T 1,q has been found as an intermediate or final topological state in the evolution of complex vortex knots (Scheeler et al 2014;Kleckner et al 2016;Xiong & Yang 2019a;Liu, Ricca & Li 2020).…”

Section: Stability Diagram Of Coiled Vortex Loopsmentioning

confidence: 94%

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Topological transition and helicity conversion of vortex knots and links

Shen

Yao²,

Hussain³

et al. 2022

J. Fluid Mech.

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Topological transition and helicity conversion of vortex torus knots and links are studied using direct numerical simulations of the incompressible Navier–Stokes equations. We find three topological transitional routes (viz. merging, reconnection and transition to turbulence) in the evolution of vortex knots and links over a range of torus aspect ratios and winding numbers. The topological transition depends not only on the initial topology but also on the initial geometry of knots/links. For small torus aspect ratios, the initially knotted or linked vortex tube rapidly merges into a vortex ring with a complete helicity conversion from the writhe and link components to the twist. For large torus aspect ratios, the vortex knot or link is untied into upper and lower coiled loops via the first vortex reconnection, with a helicity fluctuation including loss of writhe and link, and generation of twist. Then, the relatively unstable lower loop can undergo a secondary reconnection to split into multiple small vortices with a similar helicity fluctuation. Surprisingly, for moderate torus aspect ratios, the incomplete reconnection of tangled vortex loops together with strong vortex interactions triggers transition to turbulence, in which the topological helicity decomposition fails due to the breakdown of vortex core lines.

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Section: Stability Diagram Of Coiled Vortex Loopsmentioning

confidence: 94%

“…This unknotting process implies a possible route for the topological degeneration of vortex knots/links, i.e. the decrease of the topological complexity (Kleckner et al 2016;Liu et al 2020) or values of p and q of the vortex core line in the evolution of T p,q .…”

Section: Topological Degeneration For Large Torus Aspect Ratiosmentioning

confidence: 99%

Topological transition and helicity conversion of vortex knots and links

Shen

Yao²,

Hussain³

et al. 2022

J. Fluid Mech.

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“…In particular, we demonstrate that the knotting process of field lines undergoes a cascade via a sequence of reconnections of two orthogonal, counter-helical and unknotted flux tubes, as an inverse process of the unknotting cascade (e.g. Kleckner et al 2016;Liu & Ricca 2016;Liu, Ricca & Li 2020) from an initial knotted tube. Here, helical flux tubes are generally considered as a model to study the formation and evolution of solar coronal loops and sunspots (e.g.…”

Section: Introductionmentioning

confidence: 92%

Magnetic knot cascade via the stepwise reconnection of helical flux tubes

Hao

Yang

2021

J. Fluid Mech.

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“…As topological excitation, knots has relatively strong robustness; however, it is not absolutely stable, with the irreversible dissipation of energy, it will degenerate from complexity to simplicity, until mediocrity and disappear. Liu, Ricca and LI pointed out that when there is no external interference, the knot complex system will degrade and dissipate spontaneously in the way of cascade degradation 39 . The degradation process may choose different paths, among which the shortest one has the greatest probability of occurrence.…”

Section: Consciousness Field Generatormentioning

confidence: 99%

The Logic Fundamentals of Machine Consciousness: Theory of Tri-state

2022

JRAR

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For a long time, the system of scientific methodology has been composed of logic, empirical (falsification), qualitative, quantitative and deterministic, and corresponding thinking tools. However, under the background of complexity science, the category of methodology should be changed, that is, on the basis of traditional methodology, non-classical logic, hierarchy, stereotype (topological invariant) and uncertainty should be added. This is also the main idea behind the “Thoery of Tri-state” in the first part of this paper. The core idea in the theory of “Tri-state” is “Tri-state Logic” (“positive | negative | uncertain state”). The ontology of “Tri-state Logic” aims to reveal the meta space-time movement law of things transforming from one form to another, that is, the coupling of time and space in the development of things, and the orientation and evolution of the continuity of things. The mathematical basis of “Tri-state Logic” is knot theory and dynamics theory. The second part of this paper designs a machine-consciousness model framework based on the “Theory of Tri-state” (Tri-state Logic). Its research starting point is the perspective of cognitive dynamics (cognitive psychology + dynamics), which is very different from the research ideas proposed by Minsky's “The Emotion Machine”. At the same time, this paper also tries to answer Turing's questions from different space-time dimensions, and gives an experimental idea of “kindergarten game” by comparing Turing's “imitation game”.

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Minimal unlinking pathways as geodesics in knot polynomial space

Cited by 6 publications

References 26 publications

Topological transition and helicity conversion of vortex knots and links

Topological transition and helicity conversion of vortex knots and links

Magnetic knot cascade via the stepwise reconnection of helical flux tubes

The Logic Fundamentals of Machine Consciousness: Theory of Tri-state

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