Static knot energy, Hopf charge, and universal growth law

We present a family of static knotted soliton energy functionals governing the configuration maps from the Euclidean space R4n−1R4n−1 into the unit sphere S2nS2n so that the knot charges are naturally represented by the Hopf invariants in the homotopy group π4n−1(S2n)π4n−1(S2n) and the special case n=1n=1 recovers the classical Faddeev knot energy. We establish the general result that the minimum energy or the knot mass ENEN of knotted solitons of the Hopf charge N   satisfies the universal fractional-exponent growth law EN∼|N|(4n−1)/4nEN∼|N|(4n−1)/4n, in which the fractional exponent depends only on the dimensions of the domain and range spaces of the configuration maps but does not depend on the detailed structure of the knot energy.