Surfaces defined by bending of knots

Abstract

We consider the definition of the infinitesimal bending of a curve as a vector parametric equation of a surface defined by two free variables: one of them is free variable u which define curve and another is bending parameter ?. In this way, while being bent curve is deformed and moved through the space forming a surface. If infinitesimal bending field is of constant intensity, deformed curves form a ruled surface that represents a ribbon. In particular, we consider surfaces obtain by bending of knots both analytically and graphically. We pay attention to the torus knot and possibility of its infinitesimal bending so that the surface determined by bending is a part of the initial torus.