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Geometric Properties and Problems of Thick Knots
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Authors: Yuanan Diao and Claus Ernst (Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC, Department of Mathematics, Western Kentucky University, Bowling Green, KY) 
Book Description:
In geometric knot theory, a central issue is to study the various geometric properties of knots when the knots have certain thickness. This setting makes a knot more like one that is tied with a uniform physical rope. These problems are mostly motivated by the recent applications of knot theory in fields such as biology and polymer chemistry.In this book, the authors first give a brief review of the basic concepts and terminologies such as the thickness of a knot and the ropelength of a knot. They then review the main results in this field. The topics include results on the global minimum ropelength of knots, various lower and upper ropelength bounds of knots in terms of their crossing numbers, and lower and upper bounds on the total curvatures of thick knots. Some special families of knots or under different settings, such as lattice knots and smooth knots are also considered. While some proofs are omitted or only outlined due to the page limitation of the book, many important ideas, methods, and theorems are explained in depth. At the end of the book, a list of some open problems in this field is given.

Table of Contents:

1. Introduction, pp. 1-2

2. Some Basic Facts of Knot Theory, pp. 3-6

3. Thicknesses of Knots, pp. 7-12

4. Ropelengths of Knots, pp. 13-14

5. Smooth Knots vs Knots on the Cubic Lattice, pp. 15-18

6. Global Lower Bound on Ropelength of Knots, pp. 19-28

7. General Lower Bounds on Ropelength of Knots, pp. 29-32

8. General Upper Bounds on Ropelength of Knots, pp. 33-42

9. Ropelength Upper Bounds for Special Classes of Knots, pp. 43-52

10. The Spectrum of Powers Realizable by Thick Knots, pp. 53-56

11. Total Curvature of Thick Knots, pp. 57-66

12. The Linking Number of a Thick Link with Two or More
Components, pp. 67-70

13. Some Open Questions, pp. 71-72

References,pp. 73-78


      Mathematics Research Developments
   Binding: Softcover
   Pub. Date: 2009
   ISBN: 978-1-60741-070-6
   Status: AV
Status Code Description
AN Announcing
FM Formatting
PP Page Proofs
FP Final Production
EP Editorial Production
PR At Prepress
AP At Press
AV Available
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Geometric Properties and Problems of Thick Knots