1.13  Braid family representatives 

Minimum braids are defined, described, generated and presented in tables for knots up to ten crossings and oriented links up to nine crossings by T. Gittings (2004). T. Gittings used them for studying graph trees, amphicheirality, unknotting numbers and periodic tables of KLs. 

According to Alexander's Theorem (1928), any oriented KL can be represented as a closed braid. A braid representation is not unique: every KL has an infinite number of braid representations. Among them it is possible to choose a minimal one specifying a set of restrictions. This choice depends on the criteria used for defining a minimal braid. Accepting the minimal number of strands, i.e., the smallest braid index as the main criterion, we obtain minimal braids. The minimal braid is not unique, since many braids for a given KL can have the same (minimal) number of strands. 

Four restrictions that are used to define a minimum braid are given by T. Gittings (2004, Definition 1)
 

Among the set of braids for any KL, the minimum braid is the one that has the following properties: 

  1. minimum number of braid crossings; 
  2. minimum number of braid strands; 
  3. minimum braid universe
  4. minimum binary code for alternating braid crossings.
These criteria are listed in descending order of importance for determining minimum braids. 

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