By using the LinKnot function fGraphKL (webMathematica fGraphKL) you can calculate and draw the graph G of a KL given by a Conway symbol, Dowker code, or P-data. The result is the graph given by its list of unordered pairs. For that graph you can then calculate the dichromatic polynomial ZG(q,v) according to the rules mentioned.

In a vertex coloring of a graph G we use z colors to color the vertices of G, so that adjacent vertices are colored differently. The chromatic polynomial of a graph G, introduced by Birkhoff in 1912, is a function of the graph G and of the number of colors z that gives the number of different colorings of a graph G by z colors. For every graph of a KL, calculated by the LinKnot function fGraphKL we can obtain its chromatic polynomial by using the Mathematica function ChromaticPolynomial. For example, the graph of the trefoil knot is a triangle, so its chromatic polynomial is z(z-1)(z-2).

The program Knot 2000 (K2K) provides a series of functions for computing polynomial invariants of KLs from P-data or from a braid word. Depending on the type you choose, the function Skein Polynomial (webMathematica Skein Polynomial) computes the Jones polynomial, HOMFLYPT polynomial (or two-variable Jones polynomial), Alexander polynomial by the Conway relation, and the Conway polynomial. In order to compute polynomial invariants from a braid word, or from the Burau representation for a braid word, you can use the Knot 2000 functions JonesPolynomialbyBraid, AlexanderPolynomialbyBurauRep, and ThreeParallelPolynomialInvariant. The LinKnot function fAlexPoly (webMathematica fAlexPoly) calculates the multi-variable Alexander polynomial of a KL from its Conway symbol by using the Wirtinger presentation of KL. The functions KauffmanPolynomial (webMathematica KauffmanPolynomial) and RedKauffmanPolynomial compute Kauffman polynomial and reduced Kauffman polynomial of a KL given by P-data.