## The "Color Tests"

An n-color test is defined through an nxn matrix whose elements take values on {1,2,...,n}. When a color test is applied to some knot projection P, the strands of P are mapped to elements of the set {1,2,...,n}. Let i,j,k be the numbers corresponding to three strands that appear at a crossing of P, and let j correspond to the overcrossing strand, as shown in this page .

The mapping is considered successful if k=M(i,j) for all crossings of P.

Not any nxn matrix however is relevant. In order for a color test to be valid, one must make sure that ambient isotopic knots yield identical results. In other words, let P and P' be two regular projections connected through Reidemeister moves. Then each successful mapping of P should correspond to exactly one successful mapping of P'. This is necessary in order to be able to distinguish inequivalent knots.

Let K and K' be two knots possessing m and m' successful mappings for the same color test. If m and m' are unequal, one has shown that K and K' are not ambient isotopic.

The condition above implies the following constraints on M.

1. M(i,i)=i

2. M(i,j)=M(i',j) <=> i=i'

3. M(i,j)=k, M(i,l)=m, M(j,l)=n => M(m,n)=M(k,l)

In addition, one is interested in irreducible tests only; a color test is called irreducible, if there is no subset S of {1,2,...,n} other than the empty set and {1,2,...,n} itself, such that i belongs to S => M(i,j) belongs to S for all j. This is because if a test is reducible, if one strand is mapped to an element of S, then all strands are mapped to elements of S. Therefore either all or no strands are mapped to elements of S, and thus the information obtained is equivalent to information obtained from simpler tests.

Finally by permuting 1,2,...,n, a single color test may yield a number of additional tests; all such tests however yiled identical information, and are thus considered the same. Similarly two tests M and N are considered the same if M(i,j)=k <=> N(k,j)=i, since they are related through mirror symmetry.

Under these constraints, one obtains the following number of color tests.

• One color M(1,1)=1. This test however is not useful, since any knot posseses exactly one successful mapping, namely the one mapping each strand to 1. No knots can be shown inequivalent through such test.

• Three colors M(1,1)=1, M(2,1)=3, M(3,1)=2, M(1,2)=3, M(2,2)=2, M(2,3)=1, M(3,1)=2, M(3,2)=1, M(3,3)=3. This is the original 3-color test through which the trefoil's non-triviality was shown.

• Between four and eleven colors There exist 1 test for 4 colors, 2 tests for 5 colors, 2 tests for 6 colors, 3 tests for 7 colors, 2 tests for 8 colors, 6 tests for 9 colors, 1 test for 10 colors, 5 tests for 11 colors.

• More than 11 colors The time needed to run a computer program checking all possibilities becomes prohibitive. One however may generalise from tests already obtained and find such color tests. To see such an example, please proceed to the page discussing the Alexander polynomial.

Charilaos Aneziris, charilaos_aneziris@standardandpoors.com