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Total curvature, ropelength and crossing number of thick knots†

Published online by Cambridge University Press:  01 July 2007

Y. DIAO  and
C. ERNST
Y. DIAO
Affiliation:
Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC 28223, U.S.A. e-mail: ydiao@uncc.edu
C. ERNST
Affiliation:
Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, U.S.A. e-mail: claus.ernst@wku.edu
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Abstract

We first study the minimum total curvature of a knot when it is embedded on the cubic lattice. Let be a knot or link with a lattice embedding of minimum total curvature among all possible lattice embeddings of . We show that there exist positive constants c1 and c2 such that for any knot type . Furthermore we show that the powers of in the above inequalities are sharp hence cannot be improved in general. Our results and observations show that lattice embeddings with minimum total curvature are quite different from those with minimum or near minimum lattice embedding length. In addition, we discuss the relationship between minimal total curvature and minimal ropelength for a given knot type. At the end of the paper, we study the total curvatures of smooth thick knots and show that there are some essential differences between the total curvatures of smooth thick knots and lattice knots.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 1 , July 2007 , pp. 41 - 55
Copyright
Copyright © Cambridge Philosophical Society 2007

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