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(1) It is a theorem of Stallings [1978, "Constructions of fibred knots and links"] that closures of positive braids are always fibered links. Thus "most" knots are not realised as the closure of a positive braid.

(2) If you allow positive (say) crossings only, and TL generators, then you can use the latter to "rotate" the former by 90 degrees. Then the given diagram can be converted without too high a "cost".

(3) Interesting question. I'll guess that the blow up is at worst polynomial.

(1) It is a theorem of Stallings [1978, "Constructions of fibred knots and links"] that closures of positive braids are always fibered links. Thus "most" knots are not realised as the closure of a positive braid.

(2) If you allow positive (say) crossings only, and TL generators, then you can use the latter to rotate the former by 90 degrees. Then the given diagram can be converted with at most a linear growth in complexity.

(3) Interesting question. I'll guess that the blow up is at worst polynomial.

(1) It is a theorem of Stallings [1978, "Constructions of fibred knots and links"] that closures of positive braids are always fibered links. Thus "most" knots are not realised as the closure of a positive braid.

(2) If you allow positive (say) crossings only, and TL generators, then you can use the latter to "rotate" the former by 90 degrees. Then the given diagram can be converted without too high a "cost".

(3) Alexander [1923, "A lemma on systems of knotted curves"] gives an algorithm to convert a given diagram $D$ into a braid closure. I think this gives a polynomial bound (on the number of crossings after, in terms of the number of crossings before).

(1) It is a theorem of Stallings [1978, "Constructions of fibred knots and links"] that closures of positive braids are always fibered links. Thus "most" knots are not realised as the closure of a positive braid.

(2) If you allow positive (say) crossings only, and TL generators, then you can use the latter to "rotate" the former by 90 degrees. Then the given diagram can be converted without too high a "cost".

(3) Interesting question. I'll guess that the blow up is at worst polynomial.

(1) It is a theorem of Stallings [1978, "Constructions"] that closures of positive braids are always fibered links. Thus "most" knots are not realised as the closure of a positive braid.

(2) If you allow positive (say) crossings only, and TL generators, then you can use the latter to "rotate" the former by 90 degrees. Then the given diagram can be converted without too high a "cost".

(3) There is an algorithm to convert a standard diagram into a braid closure. See the "Knot Book" by Adams.

(1) It is a theorem of Stallings [1978, "Constructions of fibred knots and links"] that closures of positive braids are always fibered links. Thus "most" knots are not realised as the closure of a positive braid.

(2) If you allow positive (say) crossings only, and TL generators, then you can use the latter to "rotate" the former by 90 degrees. Then the given diagram can be converted without too high a "cost".

(3) Alexander [1923, "A lemma on systems of knotted curves"] gives an algorithm to convert a given diagram $D$ into a braid closure. I think this gives a polynomial bound (on the number of crossings after, in terms of the number of crossings before).