Energy and helicity of magnetic torus knots and braids

By considering steady magnetic fields in the shape of torus knots and unknots in ideal magnetohydrodynamics, we compute some fundamental geometric and physical properties to provide estimates for magnetic energy and helicity. By making use of an appropriate parametrization, we show that knots with dominant toroidal coils that are a good model for solar coronal loops have negligible total torsion contribution to magnetic helicity while writhing number provides a good proxy. Hence, by the algebraic definition of writhe based on crossing numbers, we show that the estimated values of writhe based on image analysis provide reliable information for the exact values of helicity. We also show that magnetic energy is linearly related to helicity, and the effect of the confinement of magnetic field can be expressed in terms of geometric information. These results can find useful application in solar and plasma physics, where braided structures are often present.