One-cusped Dehn fillings of the sisters of the Whitehead and $6^2_2$ link complements

Abstract

In this article, we investigate the arithmeticity of the one-cusped Dehn fillings of the $(-2,3,8)$-pretzel link complement and of the Berge manifold, which respectively are the sisters of the Whitehead and $6^2_2$ link complements. We show that for each such one-cusped hyperbolic Dehn filling, the cusp field, the trace field and the invariant trace field coincide. Moreover, we establish that no one-cusped hyperbolic Dehn filling of the Berge manifold is arithmetic and that the only arithmetic one-cusped hyperbolic Dehn filling of the $(-2,3,8)$-pretzel link complement is the sister of the figure eight knot complement. The techniques used to prove these results further show that each knot complement covering a one-cusped hyperbolic Dehn filling of either of these two sisters manifolds admits no hidden symmetries, effectively generalizing already known results in this regard.