If you glue opposite faces of a dedecahedron with a twist of $\frac\pi5$, you obtain the classical Poincaré sphere. It is even an integral homology sphere, and to best of my knowledge not a Seifert manifold (I am not sure, however).
But if you do the gluing with a twist of $\frac{3\pi}5$, then you obtain another homology sphere with a hyperbolic structure, so definitely not a Seifert manifold. I do not know what knot would give this manifold. But you can deduce many of its properties by drawing a picture of a dodecahedron with the appropriate corners, edges and so on identified.