Preface 
   

This interactive knot thery book provides webMathematica computations for knots and links. For all webMathematica computations, please go to the list of the LinKnot functions, or to the Appendix: new KL tables. The corresponding webMathematica LinKnot functions you can also run from the text of the book they follow.  

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Knot theory is a very inspirational field of mathematics: its basis, real knots, are familiar to everyone. Most of the basic ideas in knot theory can be formulated by using everyday language. However, it is still an area full of open questions, with more questions than answers. 

In this book we tried to accomplish a few main tasks. The first was giving links, multi-component knots, the equal role knots have by unifying them under the common name KLs (knots and links) and treating them in the same manner whenever possible. For denoting KLs was used Conway notation, a geometrical-combinatorial generic way to describe and derive KLs. That notation is also used in the Mathematica based program LinKnot. Here it works in webMathematica and represents the integral part of the book. All the functions from the LinKnot you can run also in webMathematica. All webMathematica functions are given in the list of LinKnot functions. The webMathematica computations are also provided for KLs given in the Appendix: new KL tables. LinKnot is treated not only as a following program, but as the best, extremely efficient tool for obtaining almost all results presented in the book. Each knot theoretical problem described in this book has a LinKnot function that computes the corresponding data. In this way a reader can actively use the program LinKnot, not only for illustrating some problems from the book, but as an experimental mathematical tool. Because all LinKnot functions are written in Mathematica, a reader can change them, or add some new functions, so LinKnot represents a program completely open for future development. If you have Mathematica for Windows, the program LinKnot you can download from the given address and run it in your computer.  

Beginning from shadows of KLs, we introduce graphs of KLs. The first KL invariant obtained by Component Algorithm from a KL shadow is the number of components of a KL. After introducing the relation "over"-"under" and signs of vertices in a KL shadow, we obtain KL diagrams and consider their changes, ambient isotopies and their 2D equivalents- Reidemeister moves, transforming one KL into the other. After that, we consider different notations of obtained KLs: Gauss, Dowker, and Conway notation, each of them together with their advantages and disadvantages. Other basic KL invariants such as a minimum crossing number, writhe, linking number, unknotting or unlinking number, cutting number and KL properties as chirality, periodicity or unlinking gap are explained in Chapter 1. In that chapter we also discuss the classification of KLs according to their common properties and a division of KLs into well-defined equivalence classes- families of KLs. 

In Chapter 2 two main problems of knot theory were considered: recognition and generation of KLs. In order to distinguish different KLs, as recognition criteria we consider KL colorings, KL groups, and the most powerful tool: polynomial KL invariants. Again, we try to show that polynomial KL invariants directly follow from the Conway notation and KL families. For the systematic and exhaustive derivation of KLs we accepted the concept proposed by J.H. Conway and A. Caudron, supported and used in a form adapted for computer derivation. As a prerequisite for that derivation, the complete list of basic polyhedra with n £ 20 crossings is given, as well as the list of source links derived from it. Once more, it is confirmed that most of KL properties, e.g., chirality or a number of different projections, are properties of KL families, and not merely of particular KLs. 

Chapter 3 contains a short excursion in the history of knot theory and a few non-standard applications of KLs: mirror curves, self-avoiding curves, fullerenes treated as KLs, and a possible use of KLs in mathematical logic, self-referential systems and theory of automata. 

New KL tables organized according to KL families are given in Appendix: new KL tables. The main result is that Alexander polynomials of KLs, signatures and unknotting (unlinking) numbers of all KL families generated from KLs with n 9 crossings are expressed by general formulas depending only on numbers present in a Conway notation of a KL. Moreover, a signature and unknotting (unlinking) number of every KL family is given as a linear combination of those parameters. 

The program LinKnot is primarily dedicated to an educational and experimental work with a large series of KLs (families or classes) given in Conway notation. Thanks to computers, we are now able to check large number of KLs and make some new conjectures. Therefore the program LinKnot is a tool for experimental mathematics

We are thankful to ICT and Wolfram Research for supporting our project, to Professors M. Ochiai and N. Imafuji for the cooperation in the development of the program LinKnot and for permitting its distribution together with their program Knot2000 (K2K), to Professors Donald Crowe, Louis Kauffman, Jay Kappraff, Jozef Przytycki, and Thomas Gittings for their remarks, advice and suggestions, and to all other authors that participated in the program LinKnot with the functions from their original programs. 

S. Jablan

R. Sazdanovic
 

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