The total torsion of knots under second order infinitesimal bending

In this paper we consider infinitesimal bending of the second order of curves and knots. The total torsion of the knot during the second order infinitesimal bending is discussed and expressions for the first and the second variation of the total torsion are given. Some examples aimed to illustrate infinitesimal bending of knots are shown using figures. Colors are used to illustrate torsion values at different points of bent knots and the total torsion is numerically calculated.

On torus knots and unknots

A comprehensive study of geometric and topological properties of torus knots and unknots is presented. Torus knots/unknots are particularly symmetric, closed, space curves, that wrap the surface of a mathematical torus a number of times in the longitudinal and meridian direction. By using a standard parametrization, new results on local and global properties are found. In particular, we demonstrate the existence of inflection points for a given critical aspect ratio, determine the location and prescribe the regularization condition to remove the local singularity associated with torsion. Since to first approximation total length grows linearly with the number of coils, its nondimensional counterpart is proportional to the topological crossing number of the knot type. We analyze several global geometric quantities, such as total curvature, writhing number, total torsion, and geometric ‘energies’ given by total squared curvature and torsion, in relation to knot complexity measured by the winding number. We conclude with a brief presentation of research topics, where geometric and topological information on torus knots/unknots finds useful application.

Total torsion of closed lines of curvature

In this article we investigate the total torsion of closed lines of curvature on a surface in 3 and obtain the following results.(1) The total torsion of a closed line of curvature on a surface is kπ where k is an integer. Conversely, if the total torsion of a closed curve is kπ for an integer k, then the curve can appear as a line of curvature on a surface. In particular, if the total torsion of a closed curve is 2kπ, then it can appear as a line of curvature on a closed, oriented surface of genus 1.(2) The total torsion of a closed line of curvature on an ovaloid is zero.

SPACE CURVES WITH NONZERO TORSION

For a closed curve in R3 with curvature and torsion everywhere nonzero, the sum of the total curvature and the total torsion is greater than 4π.

On the total torsion of certain nonclosed sphere curves

In this note, we establish some results concerning the total torsion and the total absolute torsion of certain non-closed stereographically projected analytic curves. The method of proof involves only elementary techniques of integration, a periodicity argument and Liouville's Theorem.