The Alexander-Conway Polynomial

As discussed in another page, there are three conditions which a matrix M(i,j) must satisfy in order to define a valid color test. To review these conditions and for a brief introduction to color tests, click here . One particular class of such tests are the so-called linear tests defined through the equation

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,

Copyright 1995


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