4 problems

Let $S^{3}\setminus K$ be the complement of a hyperbolic knot $K$ in $S^{3}$. A closed embedded totally geodesic surface is a closed totally geodesic surface embedded in this hyper…

Let $M$ be a closed orientable $3$-manifold. A TGS link is a link $K$ embedded in a $3$-manifold such that $M\setminus K$ is hyperbolic and $K$ has at least one totally geodesic sp…

Let $M_j=S^3\smallsetminus K_j$ be the complement of the twist knot $K_j$ with $j$ half-twists, and let $N$ be its thrice-punctured-sphere totally geodesic surface. For $j\geq 3$,…

Let $d$ be a positive square-free integer, let $\boldsymbol{\Gamma}_d$ denote the relevant Bianchi group, and let $\mathbb{H}^3$ be hyperbolic three-space. Define the finite set…