examples illustrating a(1) through a(7)

HOW TO READ:

  Imagine n strands flowing from left to right,
subject to periodicity: the top and bottom rows
are also considered adjacent.

  An 'X' indicates a crossing of strands; a
crossing which exchanges the top and bottom
strands has its 'X' split into '\' and '/'.

  The common property of the periodic braids below
is that every strand is at some point adjacent
to every other strand in the braid, and that the
number of crossings is minimized.

HISTORY:
  - Nov 24 2011: Initial version
  - Dec 11 2012: a(7) corrected, a(8) and a(9)
                 removed pending verification.


a(1):

       *-*

a(2):

       *-*

       *-*

a(3):

       *-*

       *-*

       *-*

a(4):

       *-*                                                                             
                                                                                       
       *-*                                                                             
                                                                                       
       * *                                                                             
        X                                                                              
       * *        

a(5):

          \                                                                            
       * * *                                                                           
        X                                                                              
       * * *                                                                           
          X                                                                            
       *-* *                                                                           
                                                                                       
       *-*-*                                                                           
                                                                                       
       *-* *                                                                           
          /     


a(6):

          \                                                                            
       * * *-*                                                                         
        X                                                                              
       * * *-*                                                                         
          X                                                                            
       *-* *-*                                                                         
                                                                                       
       * *-* *                                                                         
        X   X                                                                          
       * *-* *                                                                         
                                                                                       
       *-* *-*                                                                         
          /      

a(7):

      * *-* *-*-*
       X   X
      * *-* *-* *
               X
      *-* *-* * *
         X   X
      * * * * *-*
       X   X
      * *-* *-* *
               X
      *-* *-*-* *
         X
      *-* *-*-*-*