Differential Geometry and Topology of Curves 2024, 44: Constant Curvature in Euclidean Space and Applications

Curves with constant curvatures are described by Frenet systems of formulas, which are linear homogeneous ordinary DEs of first order with constant coefficients. In summary this system is

These can be solved to identify classes of curves. The class of trigonometric curves consists of those with position vectors whose components are trigonometric polynomials of some shared parameter. If that parameter is also the arclength, then the constant complex vectors characterizing the position vectors satisfy an easy system of equations themselves. With this, the Fenchel inequality can be eventually generalized to Cʳ-regular closed curves in more than 3D Euclidean spaces where for some integers 3 ≤ m ≤ n - 1, m + 3 ≤ r the curvatures are positive

Alexander's polynomials can be constructed for knots and links with a simpler method by abstract notation for the three different kinds of relationships of two edges. Note L₊ be a crossing of two same-directed edges where the left one crosses above the right one, L₋ the inverse of this, and L₀ two same-directed edges that don't intersect. Then, Alexander's Polynomial is

This can be even further simplified to a set of Jones' Polynomials V(t)