Zhang Fuji

It is well known that a link diagram correspond to a pair of dual signed plane graphs and the Kauffman brackets polynomial of the link can be obtained by computing the generalized Tutte Polynomial of one of its corresponding signed plane graph. Recently we extent the concept of chain polynomial to signed graph and used it to deal with the Tutte polynomials of homeomorphism of classes of signed graphs. Furthermore we define an equivalence relation on the set of link diagrams according to the homeomorphic types of their corresponding signed plane graphs. For each equivalence class we only need to compute the chain polynomial of a unique signed plane graph (say universal graph with respect to the equivalence relation ). Then the Kauffman brackets polynomial of any member of the class can easily obtained from the chain polynomial by a special parametrization.We determine the universal graph proved that the universal graphs with cyclomatic number of 1, 2, 3, 4 and 5 are 1, 1, 4, 16 and 116 respectively. In this talk we define a new equivalence relation on the set of link diagrams according to the their corresponding signed plane graphs. For each equivalence class we only need to compute the chain polynomial of a unique signed plane graph (the universal graph with respect to the new equivalence relation). Then the Kauffman brackets polynomial of any member of the class can easily obtained from the chain polynomial by a special parametrization. We determine the universal graph and proved that the universal graphs with cyclomatic number of 1, 2, 3, 4 and 5 are 1, 1, 1, 1 and 2 respectively. Finally their chain polynomial are also computed. results.