Timeline for "Effectivity" of braid notation for knots
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Aug 21, 2021 at 0:40 | comment | added | dvitek | 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. | |
| Aug 20, 2021 at 16:09 | vote | accept | Hauke Reddmann | ||
| Aug 20, 2021 at 15:45 | comment | added | dvitek | 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. | |
| Aug 20, 2021 at 14:03 | history | edited | Sam Nead | CC BY-SA 4.0 |
phrasing
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| Aug 20, 2021 at 12:35 | history | edited | Sam Nead | CC BY-SA 4.0 |
Hmmm
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| Aug 20, 2021 at 12:23 | history | edited | Sam Nead | CC BY-SA 4.0 |
added ref.
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| Aug 20, 2021 at 12:03 | history | answered | Sam Nead | CC BY-SA 4.0 |