TITLE: When do two positive braids represent the same link?
QUESTION [10 upvotes]: 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?

REPLY [4 votes]: The answer is "No". Consider two braids
$$
\beta_1 = s_1^3 s_2^3 s_3,\qquad \beta_2 = s_1 s_2^3 s_3^3
$$
They obviously represent the same knot. Moreover, they are conjugate. But
Claim 1: $\beta_2$ cannot be transformed to $\beta_1$ by successive conjugations by $s_i^{\pm 1}$ preserving positivity. I.e. you need to conjugate by longer braids. This I checked on a computer. Since there are only finitely many positive braids of given length this is a finite calculation.
Claim 2: $\beta_2$ cannot be transformed to $\beta_1$ by a sequence of Markov moves preserving positivity. This is conjectural. I checked it on a computer by going through all possible stabilizations with up to 10 strands.
On the other hand, if you allow conjugations by arbitrary short braids (=lifts of permutations) instead of only conjugations by Artin generators, the statement is true by the above cited paper "Fibered Transverse Knots and the Bennequin Bound" by John B. Etnyre, Jeremy Van Horn-Morris (https://arxiv.org/abs/0803.0758) and Lemma 6.7 from Kassel, Turaev - "Braid Groups" (thanks to Vera Vertesi for the latter reference).