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 Pdata. The result is the graph given by its list of unordered pairs. For that graph you can then calculate the dichromatic polynomial Z_{G}(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(z1)(z2). The program Knot 2000 (K2K) provides a series of functions for computing polynomial invariants of KLs from Pdata 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 twovariable 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 multivariable 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 Pdata.
