Visualization of the isometry group action on the Fomenko--Matveev--Weeks manifold

Abstract: The smallest known three-dimensional closed orientable hyperbolic ma\-ni\-fold ${\M}_1$, whose volume is equal to $0.94\dots$, was obtained independently by A. Fo\-men\-ko and S. Matveev and by J. Weeks. It is known that the isometry group of the manifold ${\M}_1$ is isomorphic to the dihedral group $\D_6$ of order $12$. The aim of the present paper is to describe the lattice of the action of the isometry group $\mbox{Isom} (\M_1)$ on the manifold $\M_1$. We obtain all orbifolds which arise as quotient spaces of $\M_1$ by the action of the subgroups of $\mbox{Isom} (\M_1)$. In particular, we describe the manifold $\M_1$ as the two-fold covering of the 3-sphere branched over the knot $9_{49}$ and as the cyclic three-fold covering of the 3-sphere branched over the two-bridge knot $5_2$.

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