The other Alexander Polynomials

As stated in a previous page, when studying a linear color test one starts with a homogeneous system of n unknowns and n equations that has the form

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,

Copyright 1995


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