A braid universe is an ordered sequence of integers, where the element i represents an unsigned crossing of the ith and (i+1)th braid strands. A braid universe becomes a braid word when to each crossing is assigned a sign +1 or -1 in the same manner as was done before with KLs. In a braid word a positive (negative) crossing of the ith and (i+1)th strands is represented by ith capital (lower case) letter. A braid is alternating if even numbered generators have the opposite sign of odd numbered generators. Therefore crossings in alternating braids have capitals for the odd letters, and lower cases for the even letters. The same convention can be taken even for non-alternating braids: a crossing is called alternating if it is capital for an odd letter and lower case for an even letter. A binary code for any braid can be generated by assigning 0 for an alternating crossing and 1 for a non-alternating crossing. With the four criteria that define a minimum braid, there is always a unique minimum for any set of braids (Gittings, 2004).
We will present another approach: after defining braid family representatives (BFRs) we will establish a correspondence between BFRs and families of KLs given in the Conway notation. For a better understanding of this correspondence, together with the standard Conway notation, a braid-modified Conway notation will be introduced and used. First we define a reduced braid word, describe a general form for all reduced braid words with s = 2 strands, generate all infinite families of two-strand braid words, and establish a correspondence between them and families of KLs given in the Conway notation. Then we consider the same problem for s ³ 3. Some applications of minimum braids (Gittings, 2004) and braid family representatives are discussed in Subsection 1.12.1. All computations are made using the program LinKnot.
We use the standard definition of a braid and description of minimum braids given by T. Gittings (2004). Instead of a...a, where a capital or lower case letter a appears p times, we write ap; p is the degree of a (p Î N). It is also possible to work with negative powers, satisfying the relationships: A-p = ap, a-p = Ap. A number of strands is denoted by s, and a length of a braid word by l.