Torus knots as Hopfions

Michikazu Kobayashi,Muneto Nitta

doi:10.1016/j.physletb.2013.12.002

Abstract

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. (P,Q)-torus knots consisting of |Q| sine-Gordon kink strings twisted P/Q 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.

Highlights

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. (P, Q)–torus knots consisting of |Q| sine-Gordon kink strings twisted P/Q times into the poloidal cycle along the toroidal cycle on a toroidal domain wall carry the Hopf charge P Q, which demonstrates that Hopfions can be further classified according to torus knot type
Stableknots were first constructed in Refs. [13, 14]. This model admits solitons having a topological charge, i.e., a Hopf charge classified by the homotopy group π3(S2) Z, which are referred to as Hopfions [15, 16]. (Un)stable Hopfions have been investigated in various physical systems such as exotic superconductors [17], ferromagnets [18], and BoseEinstein condensates [19]
In our previous paper [25], we considered the FaddeevSkyrme model with a ferromagnetic potential term, that is, a potential term quadratic in the field [26, 27] admitting two discrete vacua and a domain wall with a U (1) modulus interpolating between them [27,28,29]

Summary

Introduction

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. (P, Q)–torus knots consisting of |Q| sine-Gordon kink strings twisted P/Q times into the poloidal cycle along the toroidal cycle on a toroidal domain wall carry the Hopf charge P Q, which demonstrates that Hopfions can be further classified according to torus knot type. For the (P, Q) Hopfions, |Q| sine-Gordon kink strings appear on the toroidal domain wall, which are twisted P/Q times into the poloidal cycle along the toroidal cycle with forming (P, Q)–torus knots.

Results

Conclusion