Cutting a torus
A solid torus of revolution can be cut by n (> 0) planes into maximally

  
    
      
        
          
            (
            
              
                
                  n
                  +
                  2
                
              
              
                
                  n
                  −
                  1
                
              
            
            )
          
        
        +
        
          
            (
            
              
                
                  n
                
              
              
                
                  n
                  −
                  1
                
              
            
            )
          
        
        =
        
          
            
              1
              6
            
          
        
        (
        
          n
          
            3
          
        
        +
        3
        
          n
          
            2
          
        
        +
        8
        n
        )
      
    
    {\displaystyle {\begin{pmatrix}n+2\\n-1\end{pmatrix}}+{\begin{pmatrix}n\\n-1\end{pmatrix}}={\tfrac {1}{6}}(n^{3}+3n^{2}+8n)}
  parts.The first 11 numbers of parts, for 0 ≤ n ≤ 10 (including the case of n = 0, not covered by the above formulas), are as follows:

1, 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, ... (sequence A003600 in the OEIS).
{\displaystyle \pi _{1}(\mathbf {T} ^{2})=\pi _{1}(S^{1})\times \pi _{1}(S^{1})\cong \mathbf {Z} \times \mathbf {Z} .} Intuitively speaking, this means that a closed path that circles the torus "hole" (say, a circle that traces out a particular latitude) and then circles the torus "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly latitudinal and strictly longitudinal paths commute. An equivalent statement may be imagined as two shoelaces passing through each other, then unwinding, then rewinding. If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or through the inside like rolling a sock (with the toe cut off). Additionally, if the cylinder was made by gluing two opposite sides of a rectangle together, choosing the other two sides instead will cause the same reversal of orientation. The first homology group of the torus is isomorphic to the fundamental group (this follows from Hurewicz theorem since the fundamental group is abelian). Two-sheeted cover
The 2-torus double-covers the 2-sphere, with four ramification points. Every conformal structure on the 2-torus can be represented as a two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the Weierstrass points.