On the Total Curvature of Knots
2'n, equality holding only for plane convex curves. K. Borsuk, in 1947, extended this result to n dimensional space, and, in the same paper, conjectured that the total curvature of a knot in three dimensional space must exceed 47r. A proof of this conjecture is presented below.' In proving this proposition, use will be made of a definition, suggested by R. H. Fox, of total curvature which is applicable to any closed curve. This general definition is validated by showing that the generalized total curvature K(C) is