The Self-Linking of Torus Knots (original) (raw)

Torus knots are a widely studied class of space curves, convenient because of the surface on which they lie and the natural way in which they are defined. Costa [3], Little [6] and Romero-Fuster [8] have studied curvature and torsion properties of space curves previously. In the present work, some geometric properties of these knots are presented. The main result shows that the self-linking invariant for a (p, q) torus knot changes as the rigid geometric structure of the torus on which the curve lies is changed. It has been shown by Banchoff [1] that (p, q) curves on the flat torus in S 3 have constant self-linking invariant.