We discuss a new cluster representation for the internal energy and the specific heat of the d-dimensional (d=2,3,4) Ising model, obtained by studying the percolation mapping of an Ising model with frustrated links and performing a dual transformation. Our new representation relates the thermal operators to the topological properties of the clusters of a typical configuration of Ising percolation and is a powerful tool to study the degree of connection of the sites in such clusters. We conclude the paper showing, as an application of our representation, that the thermal critical index of the Ising model is related to the fractal dimension of the so called "cutting pairs" thus answering to a long-standing open question in Ising percolation.

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