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We introduce several methods to define the self-inductance of a single loop as the regularization of divergent integrals obtained from Neumann and Weber formulae for the mutual inductance applied to a single loop. We give generalization of these two formulae in the sense of generalized functions and study the residues.

We study an energy functional of knots, e p j ( jp > 2), that is finite valued for embedded circles and takes +∞ for circles with double points. We show that for any b ϵ R there are finitely many solid tori T 1 ,…, T m such that any knot with e p j ⩽ b can be contained in some T i in a good manner. Then we can show the existence of a minimizer of e p j in each knot type.

We define energy functionals on the space of embeddings from S1 into R 3 and show the finiteness of knot types under bounded values of those functionals.