1.12.1 KLs
and braids To work with braids in knot theory, we need to know even more than which words represent the same braid. We need to recognize isotopies of closed braids represented in algebraic language, to be able to determine when the closures of two braids represent the same oriented KL. Two braids are called Markov equivalent if their closures yield the same oriented KL. Analogously to Reidemeister moves, we need to consider Markov moves a set of moves on braids that give all equivalents of any given closed braid. In a paper published in 1935 A.A. Markov formulated the theorem, now known as the Markov theorem. That theorem, in fact, describes Markov moves and proves their necessity and sufficiency. To the already mentioned equivalence operations for open braids we need to add two more operations. The first of them is called conjugation. It is a multiplication of a braid word w by b and b^{1}, on one and the other side, resulting in the word bwb^{1} or b^{1}wb. This operation corresponds to a second Reidemeister move on the KL projection. Because none of the operations introduced till now change the number of strings in the corresponding braid, we need the next operation called stabilization. By using it, we add or delete a loop in a closed braid. This operation takes a word w describing an nstranded braid and replaces it with the word wb_{n} or wb_{n}^{1} each of them corresponding to an (n+1)stranded braid. In that case, the resulting word wb_{n} or wb_{n}^{1} corresponds to a braid with one more strand. We also accept the inverse operation, where a word of the form wb_{n} or wb_{n}^{1} is replaced with the single word w, assuming that w does not contain the letters b_{n} or b_{n}^{1}. In that case, the number of strands is decreased by 1. Pictorially, two additional Markov moves are easily understood considering a closure of a braid. A conjugation is just a trivial move on a closure of a braid since closing the braid bwb^{1} or b^{1}wb allows b^{1} to cancel out the effect of b or vice versa. In stabilization, the loop that appears after closing a braid will disappear. The proof that the five operations described are sufficient to obtain from one closed braid representation of an oriented KL any other closed braid representation of the same oriented KL is given by J. Birman (1976). In the same way as before, when we tried to find the appropriate sequence of Reidemeister moves to minimize KL projections and obtain the crossing number, we have here the same problem: we know particular moves (Markov moves), but not their order and the sequence transforming one closed braid word to the other, equivalent to it. In the language of Markov moves and braid words, the transformations shown are BaBac = BaBa =aBaBaA =aBaB. A knot or link L can be formed from an infinite number of braids. Within the set of braids from which L is formed there exist braids that have a fewest number of strings (or strands). Any such braid is called the minimal braid presentation of L (or simply minimal braid), and its number of strings is called the braid index of L. The minimal braid is not unique since many braids for a given link L can have the same number of strands.
