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Thicknesses of knots

Published online by Cambridge University Press:  01 March 1999

Y. DIAO
Affiliation:
Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223
C. ERNST
Affiliation:
Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101
E. J. JANSE VAN RENSBURG
Affiliation:
Department of Mathematics and Statistics, York University, North York, Ontario, M3J 1P3, Canada

Abstract

In this paper we define a set of radii called thickness for simple closed curves denoted by K, which are assumed to be differentiable. These radii capture a balanced view between the geometric and the topological properties of these curves. One can think of these radii as representing the thickness of a rope in space and of K as the core of the rope. Great care is taken to define our radii in order to gain freedom from small pieces with large curvature in the curve. Intuitively, this means that we tend to allow the surface of the ropes that represent the knots to deform into a non smooth surface. But as long as the radius of the rope is less than the thickness so defined, the surface of the rope will remain a two manifold and the rope (as a solid torus) can be deformed onto K via strong deformation retract. In this paper we explore basic properties of these thicknesses and discuss the relationship amongst them.

Type
Research Article
Copyright
The Cambridge Philosophical Society 1999

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