Cool, i didn't know about it. I just knew about Toby Hall program (which, for several reasons, cannot be included in sage). It would be nice to produce a picture of the train track of a braid ;)
El domingo, 9 de agosto de 2015, 14:03:09 (UTC+2), fuglede....@gmail.com escribió: > > There could of course be some point to cooking up a new implementation > from scratch, but if I were to do it, I'd probably just wrap this library > <https://github.com/jeanluct/braidlab/tree/master/extern/trains> (note > that main() in train.cpp needs a few changes to be run ad hoc, but I've > had some good use out of it already). > > Den søndag den 9. august 2015 kl. 13.02.32 UTC+2 skrev mmarco: >> >> I also spent some time trying to figure out how to implement Bestvina >> -Handel algorithm. But i am not really an expert on that, so i desisted >> after some time. If you want to work on that, i would definitely try to >> help. >> >> I am so happy to have a braid/knot theory specialist on board. >> >> El domingo, 9 de agosto de 2015, 12:55:21 (UTC+2), fuglede....@gmail.com >> escribió: >>> >>> Having a libhomfly wrapper would certainly be nice. More generally, the >>> Mathematica KnotTheory` >>> <http://katlas.org/wiki/The_Mathematica_Package_KnotTheory%60> package >>> allows, iirc, for the input of a general R-matrix and spits out braid group >>> representations. Or maybe it's just for the ones that come from quantum >>> groups of simple Lie algebras. It would be great to have that ported to >>> sage as well. >>> >>> Nonetheless, I think there could still be some value in having a >>> Kauffman bracket implementation of the Jones polynomial such as yours: >>> while the braid group representation algorithm is certainly more effective >>> in particular cases, it is not clear to me that it *always* is. In >>> particular, what I wrote tends to be rather slow for closures of long words >>> in 10 or more strands. I actually noticed the incorrect outputs in #17030 >>> when I was about to benchmark the algorithm. >>> >>> Also on my wish list is a wrapper for Toby Hall's implementation of >>> Bestvina–Handel for braids, but that's a different issue entirely. >>> >>> Den søndag den 9. august 2015 kl. 12.34.45 UTC+2 skrev mmarco: >>>> >>>> I wouldsay that the problems in your examples has to do with the >>>> existence of completely isolated trivial components. Since we make the >>>> computation with the lists of crossings, all components with no crossings >>>> involved are ignored. Besides, the first example has no crossings at all >>>> (which makes the method to fail). >>>> >>>> So maybe we should try to take care of these cases in our data >>>> structures. >>>> >>>> If you are interested in knot theory in Sage, please help us review the >>>> ticket. There are some people already looking at the code, but they would >>>> like someone that knows the theory behind it to review the mathematical >>>> correctness. >>>> >>>> Btw, in the long run, i would like to include a library[1] to compute >>>> the homfly polynomial. It is much faster than our actual methods for >>>> Alexander and Jones polynomials. But in the meantime, i think we should >>>> focus on having the basics merged. >>>> >>>> [1]http://trac.sagemath.org/ticket/18047 >>>> >>>> El jueves, 6 de agosto de 2015, 17:01:55 (UTC+2), fuglede....@gmail.com >>>> escribió: >>>>> >>>>> In fact, when matching the return values of Link.jones_polynomial() >>>>> with the one I posted, I ran into some problems for sufficiently trivial >>>>> links: >>>>> >>>>> sage: B = BraidGroup(2) >>>>> sage: b = B([]) >>>>> sage: L = Link(b) >>>>> sage: L.jones_polynomial() >>>>> ... >>>>> IndexError: list index out of range >>>>> >>>>> Likewise, it does not appear to give the expected results when it does >>>>> give results: >>>>> >>>>> sage: B = BraidGroup(8) >>>>> sage: b = B([1]) >>>>> sage: L = Link(b) >>>>> sage: L.jones_polynomial() >>>>> 1 >>>>> sage: b.jones_polynomial() >>>>> A^12 + 6*A^8 + 15*A^4 + 15/A^4 + 6/A^8 + 1/A^12 + 20 >>>>> >>>>> >>>>> >>>>> This was obtained using the version of link.py in 04facf8. >>>>> >>>>> - Søren >>>>> >>>>> >>>>> Den torsdag den 6. august 2015 kl. 15.58.42 UTC+2 skrev Amit Jamadagni: >>>>>> >>>>>> Hello Soren, >>>>>> Yeah, we have used the Kauffman's bracket decomposition for >>>>>> the construction of Jones polynomial. I am not sure (may also be not the >>>>>> right person to comment) on whether we could include this in the current >>>>>> ticket. I guess may be we could have it in the groups/braid.py as we >>>>>> have >>>>>> an implementation of Alexander polynomial which is also implemented in >>>>>> the >>>>>> ticket #17030. >>>>>> >>>>>> Thanks, >>>>>> Amit. >>>>>> >>>>>> On Thu, Aug 6, 2015 at 9:20 PM, <fuglede....@gmail.com> wrote: >>>>>> >>>>>>> Hi Amit >>>>>>> >>>>>>> Thanks for the reference; good to know that stuff is happening in >>>>>>> that regard. >>>>>>> >>>>>>> And yes, everything here is related to the braid group. Even though >>>>>>> this would create some overlap, perhaps it could be of use to have both >>>>>>> algorithms: using braid group representations, for a fixed number of >>>>>>> strands, the evaluation of the Jones polynomial of the trace closures >>>>>>> becomes polynomial in the number of crossings (as only matrix >>>>>>> multiplication is involved). From a quick look at ticket #17030, that's >>>>>>> not >>>>>>> the case for the existing implementation which appears to implement >>>>>>> Kauffman's O(2^{O(#crossings)}) algorithm (please correct me if I'm >>>>>>> wrong). >>>>>>> >>>>>>> - Søren >>>>>>> >>>>>>> Den torsdag den 6. august 2015 kl. 14.28.39 UTC+2 skrev Amit >>>>>>> Jamadagni: >>>>>>>> >>>>>>>> Hello Soren, >>>>>>>> Thanks for sharing the work. But we do have been working on >>>>>>>> Knot Theory and here is the ticket >>>>>>>> Ticket : http://trac.sagemath.org/ticket/17030 >>>>>>>> <http://www.google.com/url?q=http%3A%2F%2Ftrac.sagemath.org%2Fticket%2F17030&sa=D&sntz=1&usg=AFQjCNHtremMXOZeAA7pqdSLRqPr2yNIIg>, >>>>>>>> >>>>>>>> which is currently under review. It would be helpful if you compare >>>>>>>> the >>>>>>>> missing features as the work on calculations of Jones polynomial has >>>>>>>> been >>>>>>>> included. Also from the source, as far as I understand the >>>>>>>> representations >>>>>>>> are mainly Braid Group, but we do have supported other representations >>>>>>>> such >>>>>>>> as oriented gauss code and also planar diagram. I guess you could >>>>>>>> directly >>>>>>>> contribute to the ticket, if something is missing. >>>>>>>> >>>>>>>> Thanks, >>>>>>>> Amit. >>>>>>>> >>>>>>>> On Thu, Aug 6, 2015 at 7:30 PM, <fuglede....@gmail.com> wrote: >>>>>>>> >>>>>>>>> Hey sage-devel >>>>>>>>> >>>>>>>>> In work with Egsgaard, I ended up needing an implementation of the >>>>>>>>> Jones representations of braid groups and figured it made sense to do >>>>>>>>> it in >>>>>>>>> sage. While interesting in their own right, they also allow for >>>>>>>>> direct >>>>>>>>> calculation of the Jones polynomials of the trace closures of the >>>>>>>>> braids, >>>>>>>>> and I figured that since sage is currently rather low on quantum >>>>>>>>> topology >>>>>>>>> (and knot theory in general), that adding this to the base could be >>>>>>>>> useful >>>>>>>>> in general. >>>>>>>>> >>>>>>>>> The development guide suggests suggesting changes here before on >>>>>>>>> trac, so here you go. The source code is currently available here: >>>>>>>>> >>>>>>>>> https://github.com/fuglede/jones-representation/blob/master/curverep.sage >>>>>>>>> >>>>>>>>> - Søren >>>>>>>>> >>>>>>>>> -- >>>>>>>>> You received this message because you are subscribed to the Google >>>>>>>>> Groups "sage-devel" group. >>>>>>>>> To unsubscribe from this group and stop receiving emails from it, >>>>>>>>> send an email to sage-devel+...@googlegroups.com. >>>>>>>>> To post to this group, send email to sage-...@googlegroups.com. >>>>>>>>> Visit this group at http://groups.google.com/group/sage-devel. >>>>>>>>> For more options, visit https://groups.google.com/d/optout. >>>>>>>>> >>>>>>>> >>>>>>>> -- >>>>>>> You received this message because you are subscribed to the Google >>>>>>> Groups "sage-devel" group. >>>>>>> To unsubscribe from this group and stop receiving emails from it, >>>>>>> send an email to sage-devel+...@googlegroups.com. >>>>>>> To post to this group, send email to sage-...@googlegroups.com. >>>>>>> Visit this group at http://groups.google.com/group/sage-devel. >>>>>>> For more options, visit https://groups.google.com/d/optout. >>>>>>> >>>>>> >>>>>> -- You received this message because you are subscribed to the Google Groups "sage-devel" group. 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