Showing Knot Inequivalence

As discussed before, while the Reidemeister moves can be used to show that two knots are ambient isotopic, they cannot be used to prove knot inequivalence. This is due to the lack of an upper boundary of moves that may be needed to connect two equivalent knots.

Historically the proof of the existence of non-trivial knots occured during the 1920's through the technique of tricolorability. One would attempt to draw a regular projection of a knot by using three colors. The following rules would have to be obeyed.

One may easily show by exhausting all possibilities, that tricolorability is invariant under the reidemeister moves; if two regular projections are connected through such moves, either both are tricolorable, or none.
The trefoil is tricolorable, while the trivial knot is not; therefore the trefoil is non-trivial.

Tricolorability is just one example of the color tests which may be used to distinguish inequivalent knots. One needs to find a set of rules, such that for each regular projections P and P' differing by one Reidemeister move, to each successful coloring of P corresponds exactly one successful coloring of P'. Therefore if for the same color test, projection A may be colored in n ways while projection B in m ways, where n and m are different from each other, one has shown that A and B belong to inequivalent knots.

In order to understand the significance of the color tests, one first needs to know the concept of the knot group.

Until 1984 most techniques distinguishing inequivalent knots were based directly or indirectly on the knot groups and the color tests (see, for instance the Alexander polynomial. In 1984 Jones invented a new knot invariant, while in 1985 a number of mathematiciants almost simultaneously generalised Jones' polynomial and obtained the HOMFLYPT polynomial. More recently Akutsu and Wadati generalised these invariants and were thus able to distinguish among knots bearing identical Jones polynomials.

The new knot invariants are based not on knot groups, but on the braid closure of a knot. They are thus able to distinguish among mirror symmetric knots, which was impossible with the color tests and the Alexander polynomials. Click here for more on braids.

Charilaos Aneziris,

Copyright 1995


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