Please see the polynomial description page for up-to-date descriptions of the conventions used in KnotInfo for the Jones, Homfly, and Kauffman polynomials.

Kauffman and Homfly Polynomials

We follow some historical conventions: the Homfly polynomial is given in variables v and z (Hugh Morton), while the Kauffman polynomial is given in variables a and z (Kauffman).

Any polynomial with two variables can be thought of as a polynomial in one variable with coefficients that are polynomial in the other variable. The Homfly and Kauffman polynomials are given as semicolon-delimited list of coefficients with respect to the variable z2. Because odd powers of the variables never appear in either Homfly or Kauffman polynomials, we leave them out of the encoding. Each coefficient in turn is a space-delimited representation of a polynomial in the secondary variable: v2 or a2, respectively. The format for each polynomial is as follows:

<lowest exponent> <highest exponent> <coefficients>

For example, consider the following encoding of a two-variable polynomial:

-4 2 1 0 -2 2; -4 2 1 -2 -3 1; -2 0 -1 -1

The corresponding (Homfly) polynomial is:

(v-4)-2+2*v2) + (v-4)-2*v-2-3+v2)*z2 + (-v-2-1)*z4