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Knots in the Skyrme-Faddeev model. (English) Zbl 1133.81053

Summary: The Skyrme-Faddeev model is a modified sigma model in three-dimensional space, which has string-like topological solitons classified by the integer-valued Hopf charge. Numerical simulations are performed to compute soliton solutions for Hopf charges up to 16, with initial conditions provided by families of rational maps from the three-sphere into the complex projective line. A large number of new solutions are presented, including a variety of torus knots for a range of Hopf charges. Often these knots are only local energy minima, with the global minimum being a linked solution, but for some values of the Hopf charge they are good candidates for the global minimum energy solution. The computed energies are in agreement with Ward’s conjectured energy bound.

MSC:

81T10 Model quantum field theories
35Q51 Soliton equations
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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[1] Battye, R.A. & Sutcliffe, P.M. 1998 Knots as stable soliton solutions in a three-dimensional classical field theory. <i>Phys. Rev. Lett.</i> <b>81</b>, 4798–4801, (doi:10.1103/PhysRevLett.81.4798). · Zbl 0949.58022
[2] Battye, R.A. & Sutcliffe, P.M. 1999 Solitons, links and knots. <i>Proc. R. Soc. A</i> <b>455</b>, 4305–4331, (doi:10.1098/rspa.1999.0502). · Zbl 1153.57300
[3] Battye, R.A. & Sutcliffe, P.M. 2002 Skyrmions, fullerenes and rational maps. <i>Rev. Math. Phys.</i> <b>14</b>, 29–86, (doi:10.1142/S0129055X02001065). · Zbl 1037.81101
[4] Brieskorn, E. & Knörrer, H. 1986 Plane algebraic curves. Cambridge, MA: Birkhäuser-Verlag. · Zbl 0588.14019
[5] Faddeev, L. D. 1975 Quantization of solitons. Preprint IAS-75-QS70, Institute of Advanced Study, Princeton, NJ.
[6] Faddeev, L. & Niemi, A.J. 1997 Stable knot-like structures in classical field theory. <i>Nature</i> <b>387</b>, 58–61, (doi:10.1038/387058a0).
[7] Gladikowski, J. & Hellmund, M. 1997 Static solitons with nonzero Hopf number. <i>Phys. Rev. D</i> <b>56</b>, 5194–5199, (doi:10.1103/PhysRevD.56.5194).
[8] Hietarinta, J. & Salo, P. 1999 Faddeev–Hopf knots: dynamics of linked un-knots. <i>Phys. Lett. B</i> <b>451</b>, 60–67, (doi:10.1016/S0370-2693(99)00054-4). · Zbl 0965.57006
[9] Hietarinta, J. & Salo, P. 2000 Ground state in the Faddeev–Skyrme model. <i>Phys. Rev. D</i> <b>62</b>, 081701(R). (doi:10.1103/PhysRevD.62.081701).
[10] Kundu, A. & Rybakov, Y.P. 1982 Closed-vortex-type solitons with Hopf index. <i>J. Phys. A</i> <b>15</b>, 269–275, (doi:10.1088/0305-4470/15/1/035).
[11] Lin, F. & Yang, Y. 2004 Existence of energy minimizers as stable knotted solitons in the Faddeev model. <i>Commun. Math. Phys.</i> <b>249</b>, 273–303, (doi:10.1007/s00220-004-1110-y). · Zbl 1065.81118
[12] Piette, B.M.A.G., Schroers, B.J. & Zakrzewski, W.J. 1995 Multisolitons in a two-dimensional Skyrme model. <i>Z. Phys. C</i> <b>65</b>, 165–174, (doi:10.1007/BF01571317).
[13] Vakulenko, A.F. & Kapitanski, L.V. 1979 Stability of solitons in <i>S</i><sup>2</sup> in the nonlinear <i>{\(\sigma\)}</i>-model. <i>Dokl. Akad. Nauk USSR</i> <b>246</b>, 840.
[14] Ward, R.S. 1999 Hopf solitons on <i>S</i><sup>3</sup> and R<sup>3</sup>. <i>Nonlinearity</i> <b>12</b>, 241–246, (doi:10.1088/0951-7715/12/2/005). · Zbl 0944.58015
[15] Ward, R.S. 2000 The interaction of two Hopf solitons. <i>Phys. Lett. B</i> <b>473</b>, 291–296, (doi:10.1016/S0370-2693(99)01503-8). · Zbl 0959.81010
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