Braid theory partially coincident with knot theory
Further properties
The braid group 
  
    
      
        
          B
          
            1
          
        
      
    
    {\displaystyle B_{1}}
   is trivial, 
  
    
      
        
          B
          
            2
          
        
      
    
    {\displaystyle B_{2}}
   is the infinite cyclic group 
  
    
      
        
          Z
        
      
    
    {\displaystyle \mathbb {Z} }
  , and 
  
    
      
        
          B
          
            3
          
        
      
    
    {\displaystyle B_{3}}
   is isomorphic to the knot group of the trefoil knot – in particular, it is an infinite non-abelian group.
The n-strand braid group 
  
    
      
        
          B
          
            n
          
        
      
    
    {\displaystyle B_{n}}
   embeds as a subgroup into the 
  
    
      
        (
        n
        +
        1
        )
      
    
    {\displaystyle (n+1)}
  -strand braid group 
  
    
      
        
          B
          
            n
            +
            1
          
        
      
    
    {\displaystyle B_{n+1}}
   by adding an extra strand that does not cross any of the first n strands. The increasing union of the braid groups with all 
  
    
      
        n
        ≥
        1
      
    
    {\displaystyle n\geq 1}
   is the infinite braid group 
  
    
      
        
          B
          
            ∞
          
        
      
    
    {\displaystyle B_{\infty }}
  .
All non-identity elements of 
  
    
      
        
          B
          
            n
          
        
      
    
    {\displaystyle B_{n}}
   have infinite order; i.e., 
  
    
      
        
          B
          
            n
          
        
      
    
    {\displaystyle B_{n}}
   is torsion-free.
There is a left-invariant linear order on 
  
    
      
        
          B
          
            n
          
        
      
    
    {\displaystyle B_{n}}
   called the Dehornoy order.
For 
  
    
      
        n
        ≥
        3
      
    
    {\displaystyle n\geq 3}
  , 
  
    
      
        
          B
          
            n
          
        
      
    
    {\displaystyle B_{n}}
   contains a subgroup isomorphic to the free group on two generators.
There is a homomorphism 
  
    
      
        
          B
          
            n
          
        
        →
        
          Z
        
      
    
    {\displaystyle B_{n}\to \mathbb {Z} }
   defined by σi ↦ 1. So for instance, the braid σ2σ3σ1−1σ2σ3 is mapped to 1 + 1 − 1 + 1 + 1 = 3. This map corresponds to the abelianization of the braid group. Since σik ↦ k, then  σik is the identity if and only if  
  
    
      
        k
        =
        0
      
    
    {\displaystyle k=0}
  . This proves that the generators have infinite order.
partially overlapping with partially intersecting with partially sharing with partially matching with partially aligning with