# M\"obius invariant metrics on the space of knots

## TL;DR

This paper establishes conditions for M"obius invariant metrics on the space of knots and explores how certain knot energies can induce such metrics, facilitating the study of knot evolution within M"obius geometry.

## Contribution

It introduces a criterion for M"obius invariant weighted inner products on knot tangent spaces and links knot energies to these metrics, advancing geometric analysis of knots.

## Key findings

- Identifies conditions for M"obius invariant inner products
- Shows knot energies can generate such metrics
- Provides a framework for studying knot evolution in M"obius geometry

## 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.