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Any knot or link can be written in braid notation (with implied closure of strands). Some natural questions:

  1. Assume I don't allow inverses - only overcrossings are used as generators. Anything known about the braid index then (might as well be $\infty$, i.e., this representation doesn't always exist)?

  2. Assume I additionally allow Temperley-Lieb generators (what's that called, Turaev algebra?). Same question.

  3. For all three variants - surely the braid representation has some overhead with respect to the minimum crossing representation of a knot. How large can it get?

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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 with at most a linear growth in complexity.

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

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    $\begingroup$ Question (3) can be answered in the ordinary case by stepping through a proof of Alexander's theorem, which actually constructs a braid representative. I think you'll have a somewhat easier go of it if you look into the Yamada-Vogel algorithm; see section 2.2 of Birman-Brendle, "Braids: a survey" for a good overview. In particular I believe you can get a quadratic-in-the-crossing-number bound without working too hard. $\endgroup$
    – dvitek
    Commented Aug 20, 2021 at 15:45
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    $\begingroup$ Note that Yamada-Vogel will also work in case (2). But in case (1) the Yamada-Vogel algorithm necessarily introduces negative crossings. I don't know of any techniques that can bound the "positive braid index" or "positive braid length" of a positive knot in terms of its crossing number. $\endgroup$
    – dvitek
    Commented Aug 21, 2021 at 0:40

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