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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?

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I don't think any of the algorithms obviously preserve positivity. Vogel's algorithm (which I think is what you're referring to) does a lot of Reidemeister II's, which always destroys positivity. I think if there's any pair of incoherently oriented Seifert circles, Vogel's algorithm will give a non-positive result. That said, I don't know of an obvious counter-example. –  Ben Webster Oct 15 '10 at 3:03

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up vote 4 down vote accepted

Rudolph proved that positive links are strongly quasipositive. This paper might also be relevant, which allows one to create a braid with the same number of Seifert circles and writhe. However, Yamada's algorithm doesn't seem to preserve positivity either.

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Thank you. Yamada's paper is indeed interesting. –  Sammy Black Oct 15 '10 at 18:55

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