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Curves, Knots, and Total Curvature

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abstract
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 $2\pi$, 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 $4\pi$. 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.
subject
total curvature
knots
contributor
Evans, Charles (author)
Kuzmanovich, James (committee chair)
Howards, Hugh (committee member)
date
2010-05-05T16:11:11Z (accessioned)
2010-06-18T18:57:32Z (accessioned)
2010-05-05T16:11:11Z (available)
2010-06-18T18:57:32Z (available)
2010-05-05T16:11:11Z (issued)
degree
Mathematics (discipline)
identifier
http://hdl.handle.net/10339/14699 (uri)
language
en_US (iso)
publisher
Wake Forest University
rights
Release the entire work immediately for access worldwide. (accessRights)
title
Curves, Knots, and Total Curvature
type
Thesis

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