The shapes of the knots corresponding to the special Hopfions

Abstract

Torus knots can be constructed using the Faddeev-Skyrme model. These knots are called Hopfions, whose topology is described by the Hopf charge \(C=W_{1} W_{2} \). A string is entangled to form the knot, which is characterized by the linking number Lk, which is the sum of the twisting number Tw and writhing number Wr. In this paper, we investigate the relationships between the knot shapes and Hopfions with different values of \((W_{1},W_{2} )\). We find the knots shapes are not equivalent to the Hopfions shapes even if they have same topological charge. For Hopfions with the value of \((W_{1},W_{2} )\), the shapes of the knots change with Euler angle \(\theta \). The knots have more writhing structure when \(\theta \) is smaller. If \(W_{1} <W_{2} \) the writhing number cannot totally convert to the twisting number. If \(W_{1} >W_{2} \) the writhing number can totally convert to the twisting number.