Let $B_n$ be the braid group on $n$ strands, with the usual generators: $s_1, \ldots, s_{n-1}$ and their inverses, where $s_i$ is a positive half-twist interchanging the strands labelled $i$ and $i+1$. Markov's theorem says that two elements of the braid group have closures giving the same knot in $S^3$ when they are related by a sequence of moves, each of which is either a conjugation $\beta \to s_i^{\pm} \beta s_i^{\mp}$ or the inclusion $B_n \to B_{n+1}$ taking $\beta \to \beta s_n^{\pm}$.
Recall a braid is positive if it can be written as $s_{i_1} \cdots s_{i_k}$. Conjugation does not generally preserve positive braids, but it can: $s_1^{-1} (s_1 \beta) s_1$ is of course positive when $\beta$ is.
If two positive braids represent the same knot, can they be related by a sequence of Markov moves in which only positive braids appear?
Best Answer
I'll make some comments on this question.
First of all, two positive braids which are conjugate (equivalently represented by the same labelled oriented 2-component link when taken with the braid axis) will be equivalent by a sequence of positive braid type III Reidemeister moves. This follows from Garside's solution to the conjugacy problem in the braid group (see also Theorem 9.4.2 by Thurston).
For closures of 3-strand braids, the complete classification was given by Birman-Menasco. They show that most 3-strand braids which represent the same link are conjugate, except for braids which are braid index 1 or 2, and a special class of braids $s_1^ps_2^qs_1^rs_2$, where $p,q,r \geq 2$ (restricting to the positive case) which are equivalent to $s_1^ps_2s_1^rs_2^q$. One sees directly from their classification that the braid index 1 and 2 cases have a unique positive representative up to conjugacy. For the exceptional case, they show (or refer to a result of Fein which is given on p. 100 of Birman's book) that these exceptional cases are conjugate after a single stabilization. In fact, one may check directly that the stabilization may be taken to be positive, so this answers your question in the case of pairs of positive 3-strand braids. Also, note that these two examples are related by flypes, and the equivalence by one positive stabilization holds for any pair of braids which are locally related by the same type of flype given in Figure 1.2 (but which are not necessarily conjugate).
If the answer to your question is true, then I think it might give an efficient algorithm to test if two positive braids are equivalent. For positive braids, the Seifert genus is determined by the braid index and number of crossings. When one does several positive Markov stabilizations, one sees that in the positive conjugacy class of the stabilized braid, most braid generators will occur only once. So I think one ought to be able to get a bound on the number of stabilizations needed, which would then lead to an algorithm to tell them apart. One could try to apply the techniques of Birman-Menasco (after Bennequin) to attempt to understand your conjecture.