On Ahlfors’ Schwarzian derivative and knots

Martin Chuaqui

Vol. 231 (2007), No. 1, 51–61

DOI: 10.2140/pjm.2007.231.51

Abstract

We extend Ahlfors’ definition of the Schwarzian derivative for curves in euclidean space to include curves on arbitrary manifolds, and give applications to the classical spaces of constant curvature. We also derive in terms of the Schwarzian a sharp criterion for a closed curve in ℝ³ to be unknotted.

Keywords

Schwarzian derivative, simple curve, conformal metric, knot

Mathematical Subject Classification 2000

Primary: 53A04, 53A30

Secondary: 53A55

Milestones

Received: 26 December 2005

Revised: 5 September 2006

Accepted: 19 September 2006

Published: 1 May 2007