When do two positive braids represent the same link?

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?

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The paper arxiv.org/pdf/math/0312209.pdf seems relevant, but they only deal with positive permutation braids which close to knots. – Mark Grant May 14 '13 at 6:04
What you are seeking could be regarded as an analogue of the Tait flyping conjecture for positive braids. – Ian Agol May 14 '13 at 15:55

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).