On closed twisted curves

Abstract

Twisted curves in R m {{\mathbf {R}}^m} are those which have independent derivatives up to order m m . For plane and spatial curves the twisted condition is equivalent to never vanishing curvature and torsion respectively. We give a necessary and sufficient condition for a ( q , p ) (q,p) -curve on a torus to be twisted, and use those curves to construct closed twisted curves in R m {{\mathbf {R}}^m} for all m m .