| 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.
For viewing all images from this site you need to have installed
Java and Flash Player.
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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