Timeline for Unknotting knot diagrams by Reidemeister moves and crossing changes
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 30, 2021 at 5:31 | comment | added | Zhiyun Cheng | Since we are allowed to switch crossing points, it suffices to consider a knot projection (a knot diagram without over/undercrossing information, or equivalently, an immersed curve with finitely many self-crossings) on the plane. It is well known that there exists a sequence of (flat) Reidemeister moves which transforms any such curve to a simple closed curve such that the number of crossings is non-increasing, just as Neil mentioned. This result also holds even if one replaces the plane (or $S^2$) with oriented closed surfaces with higher genera [Hass J, Scott P (1994) Topology]. | |
| Jan 24, 2017 at 7:17 | comment | added | Hsien-Chih Chang 張顯之 | A different way to see this is that any innermost loop is either empty, or contains a bigon. As a note on the history of this technique, Steinitz, in his proof to the famous theorem about convex polyhedra, proved that any inclusionwise-minimal bigon can be reduced using type III moves. | |
| Aug 10, 2016 at 0:13 | comment | added | Ian Agol | To see that type II and III moves suffice to simplify an innermost loop, take an innermost bigon inside of this loop. Then one can show that all arcs cross from one side of this bigon to the other, and intersect each other at most once. Now one can use type III moves to "move" triangles to the boundary at which point the number of intersections between arcs in this bigon decreases after another type III move. Once there are no intersections between arcs left, type III moves can be made to empty the bigon, then a type II move applied to decrease the crossing number. | |
| Aug 9, 2016 at 16:26 | comment | added | Ian Agol | Okay, that's better, although there's still some justification needed to see that type II and III suffice .... | |
| Aug 9, 2016 at 15:56 | comment | added | Neil Hoffman | @IanAgol Thanks for pointing this out. You are right. The example you give can not be simplified monotonically using just Reidemeister II and III moves (with Reidemeister II causing the problem). You are also right that the notion of "innermost" should fix the problem. | |
| Aug 9, 2016 at 15:55 | history | edited | Neil Hoffman | CC BY-SA 3.0 |
Added an extra case to correct a mistake pointed out in the comments by Ian Agol.
|
| Aug 8, 2016 at 18:52 | comment | added | Ian Agol | I don't think that your description is quite right. I think that you need to assume that the loop is ``innermost", in order to be able to apply only Reidemeister II and III moves. In this picture, I don't think it is possible to simplify using only II and III: dl.dropboxusercontent.com/u/8592391/Loop.jpeg If the loop is innermost, then one can show that all arcs crossing through are simple, and then one can show that using II and III moves, one can isotope these arcs one at a time off of the loop, at which point one can apply the Type I move. | |
| Aug 8, 2016 at 16:53 | vote | accept | domenico fiorenza | ||
| Aug 8, 2016 at 15:40 | history | answered | Neil Hoffman | CC BY-SA 3.0 |
