Tackling Fluid Structures Complexity by the Jones Polynomial

Liu, Xin; Ricca, Renzo L.

doi:10.1016/j.piutam.2013.03.021

Cited by 5 publications

(3 citation statements)

References 9 publications

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“…Using (3.11), these correspond to (3.19a,b) Following LR12, and by using the same change of variables introduced by Liu & Ricca (2013) (by taking t = e), without loss of generality we can set where ! and ⌧ are two parameters that take into account the uncertainty (or probability) associated with writhe and twist values of some reference configuration (as in figure 7 of LR12).…”

Section: Derivation Of the Skein Relations Of The Homflypt Polynomialmentioning

confidence: 99%

On the derivation of the HOMFLYPT polynomial invariant for fluid knots

Liu

Ricca

2015

J. Fluid Mech.

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By using and extending earlier results (Liu & Ricca, J. Phys. A, vol. 45, 2012, 205501), we derive the skein relations of the HOMFLYPT polynomial for ideal fluid knots from helicity, thus providing a rigorous proof that the HOMFLYPT polynomial is a new, powerful invariant of topological fluid mechanics. Since this invariant is a two-variable polynomial, the skein relations are derived from two independent equations expressed in terms of writhe and twist contributions. Writhe is given by addition/subtraction of imaginary local paths, and twist by Dehn's surgery. HOMFLYPT then becomes a function of knot topology and field strength. For illustration we derive explicit expressions for some elementary cases and apply these results to homogeneous vortex tangles. By examining some particular examples we show how numerical implementation of the HOMFLYPT polynomial can provide new insight into fluid-mechanical behaviour of real fluid flows.

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Section: Derivation Of the Skein Relations Of The Homflypt Polynomialmentioning

confidence: 99%

On the derivation of the HOMFLYPT polynomial invariant for fluid knots

Liu

Ricca

2015

J. Fluid Mech.

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“…Knot theory is immensely interdisciplinary, with results and open questions spanning many fields of science, such as physics [1][2][3][4][5][6][7][8], quantum computation [9,10], quantum cryptography [11,12], chemistry and biology [13][14][15][16], study of every day life knotting of strands [17], and complexity theory [18][19][20][21]. A key notion in knot theory is that of a knot invariant -a quantity extracted from a knot K which changes only under topology nonpreserving knot operations, such as passing the knot through itself or cutting and recombining its strand.…”

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confidence: 99%

Evaluating the Jones polynomial with tensor networks

Meichanetzidis

Kourtis

2019

Phys. Rev. E

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We introduce tensor network contraction algorithms for the evaluation of the Jones polynomial of arbitrary knots. The value of the Jones polynomial of a knot maps to the partition function of a q-state Potts model defined as a planar graph with weighted edges that corresponds to the knot. For any integer q, we cast this partition function into tensor network form and employ fast tensor network contraction protocols to obtain the exact tensor trace, and thus the value of the Jones polynomial. By sampling random knots via a grid-walk procedure and computing the full tensor trace, we demonstrate numerically that the Jones polynomial can be evaluated in time that scales subexponentially with the number of crossings in the typical case. This allows us to evaluate the Jones polynomial of knots that are too complex to be treated with other available methods. Our results establish tensor network methods as a practical tool for the study of knots.

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“…Such fluids with vortex knots and links have been the subject of much research, and besides the circulation and the helicity, there are other topoligical invariants. For example, Liu and Ricca have studied the Jones polynomial as a dynamical invariant of an inviscid fluid [42,43,56], generalizing earlier results by Moffatt et al [48,49,50].…”

Section: Topological Considerationsmentioning

confidence: 79%

Multi-valued vortex solutions to the Schrödinger equation and radiation

Davidson

2019

Preprint

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This paper addresses the single-valued requirement for quantum wave functions when they are analytically continued in the spatial coordinates. This is particularly relevant for de Broglie-Bohm, hydrodynamic, or stochastic models of quantum mechanics where the physical basis for singlevaluedness has been questioned. It first constructs a large class of multivalued wave functions based on knotted vortex filaments familiar in fluid mechanics, and then it argues that for free particles, these systems will likely radiate electromagnetic radiation if they are charged or have multipolar moments, and only if they are single-valued will they definitely be radiationless. Thus, it is proposed that electromagnetic radiation is possibly the mechanism that causes quantum wave functions to relax to states of single-valuedness, and that multi-valued states might possibly exist in nature for transient periods of time. If true, this would be a modification to the standard quantum mechanical formalism. A prediction is made that electrons in vector Aharonov-Bohm experiments should radiate energy at a rate dependent on the solenoid's magnetic flux.

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Tackling Fluid Structures Complexity by the Jones Polynomial

Cited by 5 publications

References 9 publications

On the derivation of the HOMFLYPT polynomial invariant for fluid knots

On the derivation of the HOMFLYPT polynomial invariant for fluid knots

Evaluating the Jones polynomial with tensor networks

Multi-valued vortex solutions to the Schrödinger equation and radiation

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