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 was not allowed to change color during the same strand. By strand one means the segment joining two undercrossings.
- At each crossing the three strands that met should either be colored identically, or each of them should have a different color. One was thus not allowed to use one color for two strands and one for the third, if these three strands were to meet at the same crossing.
- All three colors had to be used.

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, charilaos_aneziris@standardandpoors.com*

**Copyright 1995**

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