The SAW Notation

In order for a notation to correspond to an actual knot, the following conditions are sufficient and necessary.

Two ambient isotopic knots are related through a series of equivalence moves. There are two kiinds of equivalence moves.

These moves are permitted only if the new notation does not violate the second condition above. One may easily check that the first condition will always be met. To see how the moves affect the knot, click here.

The toughest question to answer is how to distinguish among notations that yield topologically inequivalent knots. As far as I know, the best way to do it is to convert the closed Self-Avoiding Walk notation to an alternative notation and obtain characteristics of this notation. Usually people have projected the SAW and then went on working on the regular notations. Such a method however can be done either "visually", by drawing the knot and seeing the crossings, or through methods of Analytic Geometry. The former method is unreliable and unsuitable to be given to a computer, while the latter causes rounding errors, since the crossings are obtained by solving systems with real coefficients. Since the topological type of a knot is extremely sensitive to the addition or removal of any crossing, this method can easily yield inaccurate results.

The method I prefer is to convert the closed Self-Avoiding Walk notation to the braid word of the corresponding knot. This can be done without rounding errors and without depending on visual observation as follows. First, one performs enough equivalence moves so that all steps along some chosen axis appear at the boundary of a "sphere", while the rest of the SAW at its interior. Once these steps are removed, one obtains a braid whose closure yields the knot under consideration. It is possible now to obtain the braidings, whose product yields the braid word.

Once the braid word is obtained, one uses skein relation methods to obtain any of the relevant knot polynomials, such as the Alexander-Conway, Jones, HOMFLYPT, or any of the polynomials related to Topological Field Theories. If possible, one may also calculate the Akutsu-Wadati polynomials.

Unfortunately, while both the regular projection method and the braid word yield a series of knot invariants, no new invariants have yet emerged from the closed Self-Avoiding Walk notation. On a more practical level, classifying knots through the SAW notation is much more time consuming and less efficient than through regular projections. While for instance the first two non-trivial knots possess 3 and 4 crossings respectively, their minimal lengths as closed Self-Avoiding Walks are 24 and 30 respectively. Thus while through the regular projection method one may obtain in about 9 days the first 802 non-trivial knots, through the SAW method one needs about the same time to obtain just the first non-trivial knot.


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Charilaos Aneziris,

Copyright 1996