We give an explicit construction of complex maps whose nodal line have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, ) Lissajous figure, and are therefore a subfamily of spiral knots generalising the torus knots. We then prove that such maps exist and are in fact fibrations with appropriate choices of parameters. We describe how this may be useful in physics for creating knotted fields, in quantum mechanics, optics and generalising to rational maps with application to the Skyrme-Faddeev model. We also prove how this construction extends to maps with weakly isolated singularities.
General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms We present a general construction of divergence-free knotted vector fields from complex scalar fields, whose closed field lines encode many kinds of knots and links, including torus knots, their cables, the figure-8 knot, and its generalizations. As finite-energy physical fields, they represent initial states for fields such as the magnetic field in a plasma, or the vorticity field in a fluid. We give a systematic procedure for calculating the vector potential, starting from complex scalar functions with knotted zero filaments, thus enabling an explicit computation of the helicity of these knotted fields. The construction can be used to generate isolated knotted flux tubes, filled by knots encoded in the lines of the vector field. Lastly, we give examples of manifestly knotted vector fields with vanishing helicity. Our results provide building blocks for analytical models and simulations alike. DOI: 10.1103/PhysRevLett.117.274501 Introduction.-The idea that a physical field-such as a magnetic field-could be weaved into a knotty texture has fascinated scientists ever since Lord Kelvin conjectured that atoms were, in fact, vortex knots in the aether. Since then, topology has emerged as a key organizing principle in physics, and knottiness is being explored as a fundamental aspect of classical and quantum fluids [1][2][3][4][5][6][7][8], magnetic fields in light and plasmas [9][10][11][12][13][14][15][16][17][18][19], liquid crystals [20][21][22][23], optical fields [24,25], nonlinear field theories [26][27][28][29], wave chaos [30], and superconductors [31,32].In particular, helicity-a measure of average linking of field lines-is a conserved quantity in ideal fluids [33,34] and plasmas [35][36][37]. Helicity thus places a fundamental topological constraint on their evolution [1,10], and plays an important role in turbulent dynamo theory [38][39][40], magnetic relaxation in plasmas [41][42][43], and turbulence [44,45]. Beyond fluids and plasmas, helicity conservation leads to a natural connection between the minimum energy configurations of knotted magnetic flux tubes [10,42,46] and tight knot configurations [47,48], and tentatively with the spectrum of mass energies of glueballs in the quarkgluon plasma [49][50][51].Knotted field configurations provide a natural setting for studying helicity, but more subtlety is required to tie a knot in the lines of a vector field than in a shoelace: all the streamlines of the entire space-filling field must twist to conform to the knotted region. The difficulty of constructing knotted field configurations with controlled helicity makes it challenging to understand the role of helicity in the evolution of knotted structures [1,10,12].In this Letter, we show how to explicitly construct knotted, divergence-free vector fields with a wide range of topolo...
This paper is concerned with the dynamics and interactions of Q-balls in (1+1)dimensions. The asymptotic force between well-separated Q-balls is calculated to show that Q-balls can be attractive or repulsive depending upon their relative internal phase. An integrable model with exact multi-Q-ball solutions is investigated and found to be of use in explaining the dynamics in non-integrable theories. In particular, it is demonstrated that the dynamics of small Q-balls in a generic class of non-integrable models tends towards integrable dynamics as the charge decreases. Long-lived oscillations of a single Q-ball can also be understood in terms of a deformation of an exact breather solution in the integrable model. Finally, we show that any theory with Q-ball solutions has a dual description in which a stationary Q-ball is dual to a static kink, with an interchange of Noether and topological charges.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.