Gravitational sources induced by exotic smoothness and fermions as knot complements

In this paper we will discuss Brans conjecture that exotic smoothness serves as an additional gravitational source naturally arising from the handlebody construction of the exotic $\mathbb{R}^{4}$. We will consider the two possible classes, the large and the small exotic $\mathbb{R}^{4}$. Then we calculate the Einstein-Hilbert action for both exotic $\mathbb{R}^{4}$ to show the apearance of spinor fields. Then we discuss the physical properties of these spinor fields to relate them to fermions. Finally we identify the corresponding 3-manifolds as knot complements of hyperbolic knots, i.e. the knot complements are hyperbolic 3-manifolds with finite volume. With the help of this result we confirm the Brans conjecture for both kinds of exotic $\mathbb{R}^{4}$.