Alexander's Theorem guarantees that every oriented link is the closure of some braid. In other words, the map

$$ \displaystyle \coprod\_n \mathcal B_n\longrightarrow \{\text{ oriented links }\} $$

is surjective. One algorithm (I'm actually not sure that it's Alexander's original demonstration) involves choosing a basepoint in the complement of a diagram for the link and applying Reidemeister moves until the resulting diagram winds around the point in a consistent direction, at which point the diagram is manifestly the closure of a link.

If we begin with a positive diagram for a link, then do we necessarily obtain a positive braid?