Timeline for Jones polynomial of tangles using Temperley-Lieb algbra?
Current License: CC BY-SA 3.0
7 events
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| Feb 1, 2015 at 3:09 | history | edited | Marcel Bischoff | CC BY-SA 3.0 |
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| Dec 27, 2014 at 11:08 | comment | added | Reza Rezazadegan | Yes, sure, braids are tangles. But there are tangles which are not braids namely those poor little caps which have $n$ incoming and $n-2$ outgoing endpoints. For those one needs to define a partial trace on the algebra. | |
| Dec 24, 2014 at 17:11 | comment | added | Marcel Bischoff | I would say it has a lot to do with tangles. An element in the braid group represents a tangle and taking the trace it represents a link, see figure 1 in ams.org/journals/bull/1985-12-01/S0273-0979-1985-15304-2 | |
| Dec 24, 2014 at 11:41 | comment | added | Reza Rezazadegan | The definition of Jones polynomial from TL algebras uses a trace on the algebra which is given by closing up the element and counting the number of resulting circles. One may define a "partial trace" by closing up only two connected components of this tangle to get the homomorphism I alluded to in my question. I should verify if this gives the right answer. | |
| Dec 24, 2014 at 11:34 | comment | added | Reza Rezazadegan | Thank you, I didn't know about the sign problem. But this still doesn't answer my question. As far as I know the original definition f the Jones polynomial was given in terms of a braid group representation on Hecke algebras so it didn't have much to do with tangles. | |
| Dec 22, 2014 at 18:37 | history | edited | Marcel Bischoff | CC BY-SA 3.0 |
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| Dec 22, 2014 at 18:16 | history | answered | Marcel Bischoff | CC BY-SA 3.0 |