It's known that the Seifert–Weber space (obtained from a dodecahedron by gluing opposite faces with a 3/10 turn) is an example of a non-Haken 3-manifold. Since every closed 3-manifold is virtually Haken, I was wondering: is there was a known finite cover of the Seifert-Weber space that is Haken?



What is a finite Haken cover of the Seifert–Weber space?
This is constructed (reasonably explicitly) in John Hempel's 1982 paper.


*John Hempel*, MR 664329 [**Orientation reversing involutions and the first Betti number for finite coverings of $3$-manifolds**](http://dx.doi.org/10.1007/BF01393377), *Invent. Math.* **67** (1982), no. 1, 133--142.


