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Exotic Smoothness and Quantum Gravity

Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. Thus, the negative result of Duston (arXiv:0911.4068) was a surprise. A closer look into the semi-classical approach uncovered the implicit assumption of a close connection between geometry and smoothness structure. But both structures, geometry and smoothness, are independent of each other. In this paper we calculate the "smoothness structure" part of the path integral in quantum gravity assuming that the "sum over geometries" is already given. For that purpose we use the knot surgery of Fintushel and Stern applied to the class E(n) of elliptic surfaces. We mainly focus our attention to the K3 surfaces E(2). Then we assume that every exotic smoothness structure of the K3 surface can be generated by knot or link surgery a la Fintushel and Stern. The results are applied to the calculation of expectation values. Here we discuss the two observables, volume and Wilson loop, for the construction of an exotic 4-manifold using the knot $5_{2}$ and the Whitehead link $Wh$. By using Mostow rigidity, we obtain a topological contribution to the expectation value of the volume. Furthermore we obtain a justification of area quantization.