Curves, Knots, and Total Curvature

We present an exposition of various results dealing with the total curvature of curves in Euclidean 3-space. There are two primary results: Fenchel's theorem and the theorem of Fary and Milnor. Fenchel's theorem states that the total curvature of a simple closed curve is greater than or equal to 2π, with equality if and only if the curve is planar convex. The Fary-Milnor theorem states that the total curvature of a simple closed knotted curve is strictly greater than 4π. Several methods of proof are supplied, utilizing both curve-theoretic and surface-theoretic techniques, surveying methods from both differential and integral geometry. Related results are considered: the connection between total curvature and bridge number; an analysis of total curvature plus total torsion; a lower bound on the length of the normal indicatrix.

Grantor dc:publisher

Wake Forest University

Year dc:date.issued

2010

Author dc:creator

• Evans, Charles

• total curvature

Language dc:language.iso

en_US

Handle dc:identifier.uri

http://hdl.handle.net/10339/14699

OAI identifier oai:identifier

oai:wakespace.lib.wfu.edu:10339/14699