showed the relation to the Tutte polynomial. Specialisations At various points and lines of the -plane, the Tutte polynomial evaluates to quantities that have been studied in their own right in diverse fields of mathematics and physics. Part of the appeal of the Tutte polynomial comes from the unifying framework it provides for analysing these quantities. Chromatic polynomial At , the Tutte polynomial specialises to the chromatic polynomial, where denotes the number of connected components of G. For integer λ the value of chromatic polynomial equals the number of vertex colourings of G using a set of λ colours. It is clear that does not depend on the set of colours. What is less clear is that it is the evaluation at λ of a polynomial with integer coefficients. To see this, we observe: If G has n vertices and no edges, then . If G contains a loop (a single edge connecting a vertex to itself), then . If e is an edge which is not a loop, then The three conditions above enable us to calculate , by applying a sequence of edge deletions and contractions; but they give no guarantee that a different sequence of deletions and contractions will lead to the same value. The guarantee comes from the fact that counts something, independently of the recurrence. In particular, gives the number of acyclic orientations. Jones polynomial Along the hyperbola , the Tutte polynomial of a planar graph specialises to the Jones polynomial of an associated alternating knot. Individual points (2,1) counts the number of forests, i.e., the number of acyclic edge subsets. (1,1) counts the number of spanning forests (edge subsets without cycles and the same number of connected components as G). If the graph is connected, counts the number of spanning trees. (1,2) counts the number of spanning subgraphs (edge subsets with the same number of connected components as G). (2,0) counts the number of acyclic orientations of G. (0,2) counts the number of strongly connected orientations of G. (2,2) is the number where is the number of edges of graph G. (0,−2) If G is a 4-regular graph, then counts the number of Eulerian orientations of G. Here is the number of connected components of G. (3,3) If G is the m × n grid graph, then counts the number of ways to tile a rectangle of width 4m and height 4n with T-tetrominoes.