M(i,j) = [ t * j + (1-t) * i ] mod n

This is a valid n-color test if and only if the * Greatest Common
Divisor * GCD(n,t) = GCD(n,t-1) = 1. This implies that n must be odd,
since otherwise one of the two GCD's would be even. The 3-color test is a
linear one (n=3, l=1), while the 4-color test is not. In order to avoid
repeated versions of the same test, one requires 0 < t < n . Mirror symmetric
tests are characterised by t and t' such that t * t' = 1 mod n. In such a case
one only considers min(t,t').

All knots possess a successful mapping, in which all strands are mapped to the same element of {1,2,...,n}. Since this is in general a non-zero number mod n, this proves that the homogeneous system defined through the N equations

K = t * J + (1-t) * I

possesses non-zero solutions and thus its determinant is 0.

(N is the crossing number of a knot projection, each strand corresponds to one unknown of the system, each crossing provides one of the equations, and I, J, K, are the unknowns corresponding to the three strands meeting at each crossing).

If one sets one of the unknowns equal to 0 and removes one of the N
equations, one obtains a homogeneous system with N-1 equations and N-1
unknowns. If the determinant of this system is 0 mod n, the corresponding knot
projection possesses successful mappings that are non-trivial (not all strands
are mapped to the same number). This determinant is a polynomial of t, P(t); by
switching the unknown set to 0, or the equation removed, one may multiply this
polynomial by factors of t^m or -t^m for some integer m. By imposing the
constraint that P(1)=1 and that P(0) is finite and non-zero, one obtains a knot
invariant. This invariant is called
the **Alexander-Conway** polynomial. Through this invariant one may easily
find whether a particular knot possesses non-trivial successful mappings under
some linear test; this occurs if P(t) = 0 mod n. It was first invented by
Alexander in the 1920's, while Conway in the 1960's showed that it could also
be obtained through the so-called **skein relations**.

To summarise: let t,n natural integers such that GCD(t,n)=GCD(t-1,n)=1. Then M(i,j)=[ t * j + (1-t) * i ] mod n is a valid color test. All knots admit at least n successful mappings. Knots satisfying P(t) = 0 mod n admit at least n^2 successful mappings. If two knots have different Alexander-Conway polynomials, then there is (at least) one linear color test for which one knot admits exactly n successful mappings, while the other at least n^2; therefore these two knots are not ambient isotopic.

It is possible of course that two knots having identical Alexander-Conway
polynomials may still respond differently to linear color tests and thus be
inequivalent; to see more about this case, please read about the
**other Alexander polynomials**.

*Charilaos Aneziris, charilaos_aneziris@standardandpoors.com*

**Copyright 1995**

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