Timeline for How to motivate the skein relations?
Current License: CC BY-SA 2.5
20 events
| when toggle format | what | by | license | comment | |
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| Jul 4, 2016 at 6:51 | answer | added | Qiaochu Yuan | timeline score: 5 | |
| Jul 4, 2016 at 4:27 | answer | added | Louis H. Kauffman | timeline score: 15 | |
| Apr 2, 2013 at 21:32 | answer | added | Neil Hoffman | timeline score: 2 | |
| Jun 25, 2011 at 1:26 | history | edited | John Pardon |
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| Apr 9, 2010 at 19:37 | vote | accept | Hailong Dao | ||
| Apr 6, 2010 at 16:13 | comment | added | Daniel Moskovich | Note though that calculating the Jones polynomial via a skein relation is NP. Freedman and co-authors have some papers showing it is quantum P. I.e. it's really bad for calculating if you have some random knot with loads of crossings. | |
| Apr 6, 2010 at 14:23 | answer | added | Daniel Moskovich | timeline score: 6 | |
| Apr 5, 2010 at 22:44 | answer | added | Charlie Frohman | timeline score: 26 | |
| Apr 5, 2010 at 22:22 | comment | added | David Jordan | Dear Hailong and Ryan, thanks for the clarifications. | |
| Apr 5, 2010 at 22:00 | answer | added | Sam Nead | timeline score: 3 | |
| Apr 5, 2010 at 18:46 | answer | added | Scott Carter | timeline score: 3 | |
| Apr 5, 2010 at 18:37 | comment | added | Ryan Budney | Actually, I think my condition was over-restrictive. Find an arc on the knot that contains all the crossings of the knot and whenever there is a self-crossing of the arc, the latter crossing (in the arc's ordering) is an over-crossing. This ensures the knot is the unknot. The idea is that you can think of the missing coordinate along the knot (the height that's missing in the diagram for the knot sitting in 3-space) as increasing. | |
| Apr 5, 2010 at 18:31 | answer | added | some guy on the street | timeline score: 2 | |
| Apr 5, 2010 at 18:31 | comment | added | Hailong Dao | Thanks Ryan, that would be easier to explain to my students! | |
| Apr 5, 2010 at 18:21 | comment | added | Ryan Budney | Hailong, there is an easy way to recognise "simple" diagrammatic unknots. Check to see if there is an arc of the knot where that contains all the crossings of the knot diagram and all of them are overcrossings (on the arc). That has to be the unknot. It recognises the 1-crossing knot as an unknot, for example. This is part of the algorithm Jordan is referring to, but he does not state explicitly. | |
| Apr 5, 2010 at 18:19 | answer | added | Ryan Budney | timeline score: 8 | |
| Apr 5, 2010 at 18:15 | comment | added | some guy on the street | In fact the linked w.p. page is clear that for e.g. the HOMFLYPT invariant, simpler link diagrams are not always sufficient for computation. | |
| Apr 5, 2010 at 18:06 | comment | added | Hailong Dao | Dear David, here is something I have in mind: suppose you start with the knot with 1 crossing which is of course an unknot, but you don't know it. Then by doing skein relation, the object you get are always as complicated as what you started with. In fact, you use this to compute the invariant of the 2-component unlink, but I can't convince my student that it is natural. | |
| Apr 5, 2010 at 17:46 | comment | added | David Jordan | I don't completely understand the second sentence of the second paragraph. Are you saying "I understand the proof that skein relations, plus value at the unknot determine the invariant on all knots, but I don't find it intuitive" or are you asking for an explanation? The latter is that you can get to the unknot by flipping sufficiently many crossings, using two of the terms in the Skein relation, while the leftover term has one less crossing over all. But I suspect you know that? | |
| Apr 5, 2010 at 17:29 | history | asked | Hailong Dao | CC BY-SA 2.5 |