1. Möbius invariant metrics on the space of knots
- Author
-
Jun O'Hara
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics::Combinatorics ,Mathematics::Complex Variables ,Mathematics::General Mathematics ,Mathematics::Number Theory ,Hyperbolic geometry ,010102 general mathematics ,Algebraic geometry ,Mathematics::Geometric Topology ,01 natural sciences ,Mathematics - Geometric Topology ,Knot (unit) ,Differential geometry ,0103 physical sciences ,Tangent space ,Mathematics::Metric Geometry ,010307 mathematical physics ,Geometry and Topology ,57M25, 53A30, 57M25 ,0101 mathematics ,Invariant (mathematics) ,Projective geometry ,Mathematics - Abstract
We give a condition for a function to produce a M\"obius invariant weighted inner product on the tangent space of the space of knots, and show that some kind of M\"obius invariant knot energies can produce M\"obius invariant and parametrization invariant weighted inner products. They would give a natural way to study the evolution of knots in the framework of M\"obius geometry., Comment: 14 pages
- Published
- 2020