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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