Torus knots as Hopfions

We present a direct connection between torus knots and Hopfions by finding stable and static solutions of the extended Faddeev–Skyrme model with a ferromagnetic potential term. <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></math> -torus knots consisting of <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">|</mo><mi>Q</mi><mo stretchy="false">|</mo></math> sine-Gordon kink strings twisted <math altimg="si3.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">/</mo><mi>Q</mi></math> times into the poloidal cycle along the toroidal cycle on a toroidal domain wall carry the Hopf charge PQ , which demonstrates that Hopfions can be further classified according to torus knot type