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?

  • $\begingroup$ 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. $\endgroup$ – Ben Webster Oct 15 '10 at 3:03
  • $\begingroup$ There are positive knots which are not fibered, whereas positive braids are fibered. So one couldn't possibly put a positive non-fibered knot into positive braid position. $\endgroup$ – Ian Agol Aug 29 '17 at 3:52

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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