ABSTRACT
The impact of the Tutte polynomial derives from the richness of its properties and the wealth of combinatorial information it encodes. We illustrate the importance of the Tutte polynomial by discussing briefly some central topics in its theory.
Evaluations of the Tutte polynomial at special integer points.
Basic properties of the coefficients of the Tutte polynomial, including Crapo's β-invariant.
Evaluations of the Tutte polynomial along particular curves: the chromatic, flow and reliability polynomials.
Connections between the Tutte polynomial and the Ising, Potts and random-cluster models.
Duality and the Tutte polynomial.
Medial graphs and the Martin and circuit partition polynomials; the interlace polynomial.
Connections to knot theory, in particular the Jones polynomial.
Signed, colored and topological Tutte polynomials.
