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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