Restricting the topology of 1-cusped arithmetic 3-manifolds (Q1954146)

Arithmetic manifolds establish an important link between number theory and hyperbolic three-manifolds. There are situations where arithmetic manifolds occur, for instance the cusped hyperbolic three-manifold of minimal volume. It is also worth to notice that every closed orientable three-manifold contains an arithmetic link, i.e.~ a link whose exterior is arithmetic. This follows from by the universality of the figure eight knot proved by \textit{H.~M.~Hilden, M.~T.~Lozano} and \textit{J.~M.~Montesinos} [Topology 24, 499--504 (1985; Zbl 0582.57002)]. If one considers knots instead of links the situation is very different. \textit{A. W. Reid} proved in [J. Lond. Math. Soc., II. Ser. 43, No. 1, 171--184 (1991; Zbl 0847.57013)] that the figure eight is the only arithmetic knot in the three-sphere. In [J. Knot Theory Ramifications 11, No. 6, 903--920 (2002; Zbl 1023.57008)] the authors proved that certain lens spaces do not contain any arithmetic knot. The paper under review continues this research, more precisely on the (non-)existence of arithmetic knots in closed orientable three-manifolds with finite fundamental group or with trivial first homology groups, namely spherical manifolds and integral homology spheres respectively. A three-orbifold is arithmetic if and only if it is commensurable to a Bianchi orbifold, \(\mathbb H^3/PSL(2,\mathcal O_d)\), where \({\mathcal O}_d\) is the ring of integers of the imaginary quadratic extension \( \mathbb Q(\sqrt{-d})\), for \(d\) a square-free natural number. The paper under review provides some restrictions to the quadratic field \( \mathbb Q(\sqrt{-d})\) for arithmetic knots in spherical manifolds and integral homology spheres. The main result of this paper is as follows. For an arithmetic knot in an integral homology three-sphere, the only possible values for \(d\) are \(d=1\) and \(d=3\). In addition, for arithmetic knots in spherical manifolds whose exterior covers the Bianchi orbifold, the only possibility is \(d=1\). The proof of both results starts by reducing to a finite list of Bianchi orbifolds. In the spherical case, the parabolic elements that correspond to surgery curves are restricted by using the 6-theorem, and the analysis is completed with Dunbar's list of spherical orbifolds. Different ideas for the reduction are required for integral homology spheres, where the fact that the homology of the exterior is cyclic is used. The authors mention in the introduction that the only known examples of integral homology three-spheres that contain arithmetic knots are \(1/n\)-Dehn fillings on the figure eight knot.