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One answer to your question comes from the paper The Weber-Seifert dodecahedral space is non-Haken by Burton, Rubinstein, and Tillmann.

An earlier and "easier" example is the $(1, 2)$-Dehn filling of the figure-eight knot. This manifold is hyperbolic (4.22) and non-Haken (4.41). The references are page numbers in Thurston's lecture notes.

Finally, we can use SnapPy to find presentations of the fundamental groups.

Before filling:

In[1]: M = Manifold("4_1")
In[2]: M.fundamental_group()
Out[2]: 
Generators:
   a,b
Relators:
   abbbaBAAB

After filling:

In[3]: M.dehn_fill((1, 2))
In[4]: M.fundamental_group()
Out[4]: 
Generators:
   a,b
Relators:
   abbbaBAAB
   abAbaBabAbaBAB

One answer to your question comes from the paper The Weber-Seifert dodecahedral space is non-Haken by Burton, Rubinstein, and Tillmann.

An earlier example is (say) the $(1, 2)$-Dehn filling of the figure-eight knot. This manifold is hyperbolic (4.22) and non-Haken (4.41). The references are page numbers in Thurston's lecture notes. We can use SnapPy to find presentations of the fundamental groups.

Before filling:

In[1]: M = Manifold("4_1")
In[2]: M.fundamental_group()
Out[2]: 
Generators:
   a,b
Relators:
   abbbaBAAB

After filling:

In[3]: M.dehn_fill((1, 2))
In[4]: M.fundamental_group()
Out[4]: 
Generators:
   a,b
Relators:
   abbbaBAAB
   abAbaBabAbaBAB

Edit: I remembered that Regina has the Weber-Seifert manifold as one of its examples. So, using the triangulation isomorphism signature, we can import this to SnapPy and find a presentation for its fundamental group. (We can also compute a presentation using Regina - but I prefer the notation for relations used by SnapPy).

In[5]: WS = Manifold("xvLvvvwMvQPPQQQQcehpjtqksntrtvoupwpsuwsvwcgacalvucahatbhapaggjgfk")
In[6]: WS.fundamental_group()
Out[6]: 
Generators:
   a,b,c,d
Relators:
   aDcbcdabdbbc
   aDabCADCbc
   abdbcdabdACBC
   aDBcdaDabbc
In[7]: WS.volume()
Out[7]: 11.1990647408
In[8]: WS.homology()
Out[8]: Z/5 + Z/5 + Z/5

The homology has rank three, so at least three generators are needed. It seems to be open (?) to compute the minimal number of generators.

One answer to your question comes from the paper The Weber-Seifert dodecahedral space is non-Haken by Burton, Rubinstein, and Tillmann.

An earlier and "easier" example is the $(1, 2)$-Dehn filling of the figure-eight knot. This manifold is hyperbolic (4.22) and non-Haken (4.41). The references are page numbers in Thurston's lecture notes.

Finally, we can use SnapPy to find presentations of the fundamental group before and after filling.

Before:

In[1]: M = Manifold("4_1")
In[2]: M.fundamental_group()
Out[2]: 
Generators:
   a,b
Relators:
   abbbaBAAB

After:

In[3]: M.dehn_fill((1, 2))
In[4]: M.fundamental_group()
Out[4]: 
Generators:
   a,b
Relators:
   abbbaBAAB
   abAbaBabAbaBAB

One answer to your question comes from the paper The Weber-Seifert dodecahedral space is non-Haken by Burton, Rubinstein, and Tillmann.

An earlier and "easier" example is the $(1, 2)$-Dehn filling of the figure-eight knot. This manifold is hyperbolic (4.22) and non-Haken (4.41). The references are page numbers in Thurston's lecture notes.

Finally, we can use SnapPy to find presentations of the fundamental groups.

Before filling:

In[1]: M = Manifold("4_1")
In[2]: M.fundamental_group()
Out[2]: 
Generators:
   a,b
Relators:
   abbbaBAAB

After filling:

In[3]: M.dehn_fill((1, 2))
In[4]: M.fundamental_group()
Out[4]: 
Generators:
   a,b
Relators:
   abbbaBAAB
   abAbaBabAbaBAB

One answer to your question comes from the paper The Weber-Seifert dodecahedral space is non-Haken by Burton, Rubinstein, and Tillmann.

An earlier and "easier" example is the $(1, 2)$-Dehn filling of the figure-eight knot. That this is hyperbolic and non-Haken is due to Thurston - see Chapter Four of the notes. Finally, here is a presentation of the fundamental group, obtained from SnapPy.

In[1]: M = Manifold("4_1")
In[2]: M.dehn_fill((1, 2))
In[3]: M.volume()
Out[3]: 1.39850888415
In[4]: M.fundamental_group()
Out[4]: 
Generators:
   a,b
Relators:
   abbbaBAAB
   abAbaBabAbaBAB

One answer to your question comes from the paper The Weber-Seifert dodecahedral space is non-Haken by Burton, Rubinstein, and Tillmann.

An earlier and "easier" example is the $(1, 2)$-Dehn filling of the figure-eight knot. This manifold is hyperbolic (4.22) and non-Haken (4.41). The references are page numbers in Thurston's lecture notes.

Finally, we can use SnapPy to find presentations of the fundamental group before and after filling.

Before:

In[1]: M = Manifold("4_1")
In[2]: M.fundamental_group()
Out[2]: 
Generators:
   a,b
Relators:
   abbbaBAAB

After:

In[3]: M.dehn_fill((1, 2))
In[4]: M.fundamental_group()
Out[4]: 
Generators:
   a,b
Relators:
   abbbaBAAB
   abAbaBabAbaBAB