Is there an algorithm to detect the Non-Haken Manifold?
Or, is there a sufficient condition for a manifold to be 
a non-Haken manifold? (off course, I hope that condition is not the ones in its 
definition.)


Note: a Non-Haken manifold means that either it is reducible or it contains no 2-sided properly embedded
incompressible surface.



sufficient conditions on Non-Haken manifolds
Yes, there is an [algorithm due to Jaco-Oertel](http://www.ams.org/mathscinet-getitem?mr=744850) to detect if an irreducible manifold is Haken, and therefore to detect a non-Haken manifold (if it is given to be irreducible). The phrasing is a bit confusing (are you assuming the given manifold is irreducible?), but given [Rubinstein's solution to sphere recognition](http://www.ams.org/mathscinet-getitem?mr=1403961), there is also an algorithm to detect if a manifold is irreducible. See [Jaco and Tollefson's exposition](http://www.ams.org/mathscinet-getitem?mr=1339832) for improvements to these algorithms, and the papers of Ben Burton ([arXiv](https://arxiv.org/search/math?searchtype=author&query=Burton%2C+B+A), [MathSciNet](https://mathscinet.ams.org/mathscinet/search/publications.html?pg1=INDI&s1=739103)) for implementations.


**Addendum:** Sufficient conditions for being Haken are easier than for being non-Haken. For example, if a manifold has $b\_1>0$, or has positive-dimensional $SL\_2\mathbb{C}$ character variety, then it is Haken. In fact, verifying a manifold is Haken is likely NP. However, showing something is non-Haken is more difficult. There are also techniques to analyze when a group has no fixed-point free action on a tree. Some version of this was [implemented by Fenley](https://arxiv.org/abs/math/0210482) (*Laminar free hyperbolic 3-manifolds*, Comment. Math. Helv. **82** (2007), no. 2, 247–321) to find laminar-free 3-manifolds, and in principle could be used to prove that a 3-manifold is non-Haken. However, it seems that this requires an exponential search.


Work of Dani Wise implies that a compact 3-manifold is Haken or reducible if and only if it surjects an infinite virtually-free group.


