KNOT THEORY, PARTITION FUNCTION AND FRACTALS

In this paper we first provide the open chain and the closed chain method to calculate the partition functions of the typical fractal lattices, i.e. a special kind of Sierpinski carpets(SC) and the triangular Sierpinski gaskets(SG). We then apply knot theory to fractal lattices by changing lattice graphs into link diagrams according to the interaction models, and explicitly obtain the partition functions of a special SC for the edge interaction models. These partition functions are also the knot invariants of the corresponding link diagrams. This is the first time that topology enters into fractals.