Timeline for Who thought that the Alexander polynomial was the only knot invariant of its kind?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Jun 30, 2011 at 15:29 | comment | added | Ian Agol | @Sammy: I think this paper might be considered a geometric interpretation of the Jones polynomial. front.math.ucdavis.edu/1106.4789 | |
| Apr 1, 2010 at 7:49 | comment | added | Daniel Moskovich | @Paul Regarding the second part of your comment, I don't know whether I would completely agree with the characterization of quantum topology as being about constructing invariants. I would say that what we're trying to do is to understand the structure of the set of knot invariants (at least of a certain class), and doing so entails first knowing what they all are, at least in some weak sense. This fits into the idea of understanding a space (in this case, the space of knots, or the space of 3-manifolds) by understanding the space of functions from that space to some simple target space. | |
| Apr 1, 2010 at 2:32 | comment | added | Paul | It's not completely fair to say that people had trouble constructing interesting invariants using homology since most of our understanding of high dimensional topology is obtained from invariants derived from homology, as is a lot of what we know in the classical dimension despite the zoo of QI. (eg high dimensional knot concordance and large classes of high dimensional knots were classified by homological invariants.) Before Jones, knot theory was not about constructing invariants. Now, a big part of it is. That was the change instigated by Jones, then HOMFLY/Witten/Reshetikin-Turaev | |
| Apr 1, 2010 at 1:17 | comment | added | Steven Sivek | It's probably better to say Conway rediscovered the skein relation -- it's at the end of section 12 ("Miscellaneous theorems") of Alexander's original paper, but apparently nobody noticed this for decades. | |
| Apr 1, 2010 at 0:51 | comment | added | Sammy Black | And now there is a reverse trend as well. We would like to see the Jones polynomial realized via some construction that has a more geometric flavor (that doesn't involve knot diagrams). Does this goal have any satisfactory resolution yet? | |
| Apr 1, 2010 at 0:46 | vote | accept | Sammy Black | ||
| Apr 1, 2010 at 0:43 | history | answered | Dylan Thurston | CC BY-SA 2.5 |