Timeline for Can knot diagrams be monotonically simplified using under moves?
Current License: CC BY-SA 3.0
20 events
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| Oct 26, 2021 at 14:34 | comment | added | Akiva Weinberger | Just now realizing that these under moves generalize all three Reidemiester moves. So the under move generates knot equivalence all by itself. | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Nov 28, 2014 at 14:33 | answer | added | Joel Hass | timeline score: 11 | |
| Nov 4, 2014 at 14:15 | answer | added | Bruno Martelli | timeline score: 18 | |
| Nov 3, 2014 at 19:18 | comment | added | Jesse C. McKeown | Just for fun, the unknot here has a tangle diagram with a cycle of bigons, which suggests a sequence of flypes that unknot it rather quickly. | |
| Oct 29, 2014 at 13:22 | comment | added | Dylan Thurston | Yes, there have been several examples above where you need level moves, including the Goeritz knot in the original post. Ochiai's example looks nice and simple. | |
| Oct 27, 2014 at 10:10 | comment | added | Daniel Moskovich | Ochiai's unknot is an example in which strict decreasing moves do not improve things (unless you allow also level moves): en.wikipedia.org/wiki/Unknotting_problem#mediaviewer/… | |
| Oct 9, 2014 at 14:18 | comment | added | Dylan Thurston | @IanAgol, it's easy to look for strict decreasing moves (consider each understrand, and do a shortest-path algorithm to see if it improves things). But I don't think that does work. It would still be interesting, however; for instance, it would give bounds on the total number of crossings you need to introduce using Reidemeister moves. | |
| Oct 9, 2014 at 14:02 | comment | added | Ian Agol | Gordon gave them to me a long time ago when I was a grad student thinking about grid diagrams of unknots. I think Doyle told me that Conway had a belief that one could untie an unknot using the sorts of moves you describe. I'm not sure this question ever got in the literature, but I suspect it is still open. If the strict decreasing moves worked, it would probably give a polynomial unknotting algorithm. But it's not so clear if there is only monotonic simplification. | |
| Oct 9, 2014 at 13:09 | comment | added | Dylan Thurston | @IanAgol, Marco, I also see an under move that does a similar thing, with a horizontal strand near the upper-left. Ian, where do you get these great pictures? | |
| Oct 9, 2014 at 7:22 | comment | added | Marco Golla | @IanAgol: If I don't misinterpret the diagram, there is an over move that reduces the number of crossings by 2: look at the rightmost upper corner and take that whole horizontal strand and slide it down to the bottom. If you're careful, you go from crossing four double-strands to crossing three. | |
| Oct 9, 2014 at 5:17 | comment | added | Ian Agol | What about this other one of Haken? dl.dropboxusercontent.com/u/8592391/public_html/… | |
| Oct 9, 2014 at 3:56 | history | edited | Dylan Thurston | CC BY-SA 3.0 |
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| Oct 9, 2014 at 3:55 | comment | added | Dylan Thurston | Neil, yes, that's what I intended. | |
| Oct 9, 2014 at 3:55 | comment | added | Dylan Thurston | Joe, that more general question is basically Gowers' question in the earlier question I linked to. Basically the answer is no one has one for knot diagrams, but grid diagrams provide an alternate setting where the answer is yes. (But you also contributed to that discussion, so I guess you know this.) | |
| Oct 9, 2014 at 3:53 | comment | added | Dylan Thurston | Ryan, I don't see how Whitehead doubling would make the under move obviously insufficient. If I have two parallel strands and I want to move them around, I can do under moves on them one at a time, no? | |
| Oct 9, 2014 at 1:20 | comment | added | Neil Hoffman | Just to clarify, by "monotonically simplified" is your complexity function for knot diagrams crossing number? | |
| Oct 8, 2014 at 23:00 | comment | added | Joseph O'Rourke | A more general question: Is there some (natural) set of moves that can monotonically simplify any knot? | |
| Oct 8, 2014 at 22:36 | comment | added | Ryan Budney | I'd imagine you could ensure over/under moves are insufficient by doing something like a Whitehead doubling operation on your original knot. Or a cabling. | |
| Oct 8, 2014 at 21:39 | history | asked | Dylan Thurston | CC BY-SA 3.0 |
