The Problem of Knot Classification

The reader is supposed to be already familiar with the basic notions of Knot Theory. For a brief introduction, click here.

While tremendous progress has been achieved in the study of knots, the main problem, which is to obtain a method to classify all topologically inequivalent knots, is still unsolved. During the last few years I have begun developing a series of computer programs whose aim is to provide such a solution. I personally believe that the problem has been solved in principle, although in practice, due to computer memory and time constraints, I have only been able to obtain a list with the first 250 distinct knots. In the next few pages I will explain the main ideas that are relevant to the problem. First, I will list the computer programs that are used in this project.

  1. A program that fully classifies all distinct regular knot projections.

  2. A program that identifies distinct projections belonging to ambient isotopic knots through the use of Reidemeister moves.

  3. A program that calculates the Alexander polynomials of a regular knot projection.

  4. A program that obtains valid color tests.

  5. A program that applies the color tests to regular projections.

The main idea is the following. First one obtains all distinct regular projections. Then, among projections shown to belong to ambient isotopic knots, we establish a method to keep just one of them and ignore the others. It is possible however that some of the projections that are kept do belong to ambient isotopic knots, but the previous program failed to identify them (in one of the next pages we shall show why this may occur). Therefore we calculate their Alexander polynomials. Any projections with distinct polynomials belong to inequivalent knots. In case there are projections with identical polynomials, one first obtains and then applies the color tests.

Ideally, any two projections should either be shown to belong to ambient isotopic knots through the use of Reidemeister moves, or to inequivalent knots through the Alexander polynomials and/or the color tests. This in fact would be the case if one could use infinite computer memory and wait for an infinite amount of time.

Before arriving to this point and discussing the difficulties involved, we would first like to explain the notation involved, how are knot projections converted to numbers and how are the Reidemeister moves and the color tests converted into number operations. Here you may read about the appropriate notation.

Readers that may wish to omit the discussion and go directly to the results should click here.

Charilaos Aneziris,

Copyright 1995


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