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

The determinant of such a system is always 0 and thus non-zero solutions always exist. By removing one equation and setting one unknown equal to 0 one obtains the Alexander-Conway polynomial. If for some t, P(t) = 0 mod n, one is guranteed of at least n^2 successful mappings for the corresponding color test.

The secondary Alexander polynomial is defined as follows. By removing
a second equation and setting a second unknown equal to 0, one obtains an
N-2 * N-2 determinant. The Greatest Common Divisor of all such determinants
and the original N-1 * N-1 determinant is the ** second Alexander polynomial
**. In general it is equal to a constant 1; the first knot whose second
Alexander polynomial is non-trivial has crossing number 8. If both the
Alexander-Conway and the secondary Alexander polynomial become 0 mod n for
some t and n, the knot admits at least n^3 successful mappings for the
corresponding color test. It is therefore possible to distinguish between two
knots with identical Alexander-Conway polynomials by using the secondary
Alexander polynomial.

Similarly one may proceed with the tertiary etc. Alexander polynomial, until one arrives to a polynomial equal to 1.

It is interesting to note that although an infinite number of linear color tests exists, one may obtain their information through a finite procedure by calculating the Alexander polynomials. Not more such polynomials than the crossing number can be non-trivial. Knots possessing identical all such polynomials cannot be distinguished through linear color tests.

The Alexander polynomials and the linear color tests are good enough for
knots up to 8 crossings. All distinct knots with up to 8 crossings can be
shown to be inequivalent through these methods. Among the 9 crossing knots one
finds for the first time cases of distinct knots with identical Alexander
polynomials. Such knots are shown distinct through the
**non-linear color tests**.

In case you wish to have a look at an example to see how these methods work,
click **here**. To learn about the more recently
discovered **homflypt** polynomial, click **here**
.

*Charilaos Aneziris, charilaos_aneziris@standardandpoors.com*

**Copyright 1995**

Educational institutions are encouraged to reproduce and distribute these materials for educational use free of charge as long as credit and notification are provided. For any other purpose except educational, such as commercial etc, use of these materials is prohibited without prior written permission.