Narrow Results

A 2-universal knot is a universal knot having all branching indices 1 or 2. Here we construct the first example of a 2-universal knot which could be a hyperbolic knot (SnapPea). Hence, we could have an example of a universal group generated by 180° rotations.

The main result of this paper is the construction of two Hyperbolic manifolds, $M_1$ and $M_2$, with several remarkable properties:

• ⁽¹⁾Every closed orientable $3$-manifold is homeomorphic to the quotient space of the action of a group of order $16$ on some covering space of $M_1$ or $M_2$.
• ⁽²⁾$M_1$ and $M_2$ are tesselated by 16 dodecahedra such that the pentagonal faces of the dodecahedra fit together in a certain way.
• ⁽³⁾There are 12 closed non-orientable hyperbolic surfaces of Euler characteristic $-2$ each of which is tesselated by regular right angled pentagons and embedded in $M_1$ or $M_2$. The union of the pentagonal faces of the tesselating dodecahedra equals the union of the 12 images of the embedded surfaces of Euler characteristic $-2$.