Timeline for Knot diagrams, sets of moves and equivalence relations
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 20, 2023 at 8:31 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
|
| May 18, 2015 at 23:09 | vote | accept | David Feldman | ||
| Aug 7, 2012 at 15:42 | answer | added | Ian Agol | timeline score: 8 | |
| Aug 7, 2012 at 13:07 | comment | added | Daniel Moskovich | David Feldman and Qiaochu: I think you're both right. The answer is 2-dimensional algebra, which does not suffer from imposed directionality. See the work of Dror Bar-Natan, on "the circuit algebra of tangles" and related 2-dimensional algebras. This is indeed what is happening- the abstraction is to certain diagrammatic algebras (over a "modular operad") in the sense Qiaochu is refering to. And I believe this (intentionally leaving what I mean by "this" slightly vague) is very much the appropriate abstraction. | |
| Aug 7, 2012 at 13:02 | history | edited | Daniel Moskovich |
retag knots-> knot theory
|
|
| Aug 7, 2012 at 12:38 | comment | added | Qiaochu Yuan | @David: I'm not sure what you mean by imposed directionality or why this is an issue. | |
| Aug 7, 2012 at 11:27 | answer | added | Marco Golla | timeline score: 7 | |
| Aug 7, 2012 at 11:00 | answer | added | Douglas Zare | timeline score: 6 | |
| Aug 7, 2012 at 9:41 | answer | added | Daniel Moskovich | timeline score: 14 | |
| Aug 7, 2012 at 8:09 | comment | added | David Feldman | @Qiaochu Yuan So a knot or link then belongs to Hom(0,0)? But can't a "move" fail to respect the imposed directionality? Can't it wind back and forth in "time," using perhaps some but not all of the strands at a given "time" and thus escape this optic? | |
| Aug 7, 2012 at 5:44 | comment | added | Ryan Budney | Kawauchi's survey of knot theory book mentions several of the theorems Jim Conant alludes to. | |
| Aug 7, 2012 at 3:13 | comment | added | Jim Conant | There are lots of such moves. To give an example I am very familiar with, $C_k$-moves are certain moves (surgery along claspers) that generate the equivalence relation that two knots share Vassiliev invariants of order up to k-1. Another example: Lou Kauffman first observed that two knots have the same Arf invariant iff they differ by a sequence of "band-pass" moves. | |
| Aug 7, 2012 at 3:02 | comment | added | Qiaochu Yuan | The appropriate abstraction here seems to me to be a category presented by generators and relations. (The category relevant to knots is the tangle category: math.ucr.edu/home/baez/tangles.html) This is a very general construction: it has as special cases monoids presented by generators and relations, as well as posets... | |
| Aug 7, 2012 at 2:56 | history | asked | David Feldman | CC BY-SA 3.0 |