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Hyperbolic 3-Manifolds as Discretized Configuration Spaces of Simple Graphs
\\ By Michelle Chu
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A discretized configuration space is a topological space of possible configurations of particles in which multiple particles are not allowed in neighbouring edges. In this paper, we consider discretized configuration spaces of three particles on 1-dimensional simple graphs, in particular, the discretized configuration spaces $D_3(K_7)$ and $D_3(K_{4,4})$. We prove that in both cases, removing certain vertices from the discretized configuration spaces on three particles results in complete finite-volume hyperbolic 3-manifolds. We describe their construction and triangulations by cubes and tetrahedra. We also discuss their commensurability class in relation to each other and to the complement of the figure-8 knot. 
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