Nice work of documentation. Here is some further documentation that could be interesting:
About methods to compute alexander polynomials, the classical result is Theorem 3.11 from Birman's book (note that this is basically what you said, since the burau representation is deeply related to the Fox derivatives of the relations given by the action of the braid on the free group): http://books.google.de/books?id=thv7L4AQ3J4C&printsec=frontcover&hl=es&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false This is an easy introduction to three different methods to compute Alexander Polynomials: https://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http%3A%2F%2Fwww.ucl.ac.uk%2F~ucbpeal%2Falexandermac.pdf&ei=9uQNU_fLA4mHtAa7roGoBQ&usg=AFQjCNEQEoaWGCkv1MhFLuz7qADnJ29zQg&sig2=cVCQv8y0eSRh_cryPaSKHQ&bvm=bv.61965928,d.Yms There is plenty of documentation for other invariants too. Note also that there are plenty of external software that can do computations with knots and links. It would also be interesting to study the possibility of wrapping them instead of rewriting the methods. It is true that some of them are old and no longer mantained, but it might be worth taking a look nevertheless. Some links for this: http://www.math.kobe-u.ac.jp/~kodama/knot.html http://burtleburtle.net/bob/knot/homfly.html http://www.math.uic.edu/t3m/ http://www.liv.ac.uk/~su14/knotprogs.html Also it could be possible to use some method based on Skein relations, but i am not sure how good it would work. El martes, 25 de febrero de 2014 22:38:20 UTC+1, Amit Jamadagni escribió: > > I have been going through the implementation Knot Atlas, as per my > understanding they have stored the knots in a table and have inputted it to > get various results, I would like to know whether we will be using tables > to input or any other way to input. I have seen through Vogel's algorithm > which takes in the oriented gauss code and ends up giving out the braid > word [1] from which we can construct the knot(as it is closed braid) and > from the braid word we can calculate the Alexander polynomial from the > Burau's representation(which is currently implemented in braid in Sage) (I > have heard is not true for strings greater than 5). I have been reading > through [2] which mentions how to calculate Seifert matrix from a braid > representation which can be again used to represent knots. I have been > searching for algorithms which would relate the Seifert matrix and various > invariants. It would be really helpful if I could get more reference to the > algorithms that could be implemented. I have just started the > implementation details mentioned in [3]. > > [1] http://magma.maths.usyd.edu.au/~danr/site/talks/20070531.pdf > [2] http://www.maths.ed.ac.uk/~jcollins/SeifertMatrix/SeifertMatrix.pdf > [3] http://www.layer8.co.uk/maths/braids/braid-user-documentation.html > > On Tue, Feb 25, 2014 at 7:16 PM, kcrisman <kcri...@gmail.com <javascript:> > > wrote: > >> >>> >>> [2] http://legacy.earlham.edu/~peters/knotlink.htm#software >>> >>> >> *"June 2, 2004*. Unfortunately I no longer have time to update *Knots on >> the Web*. I know it conntains many dead links and omits many good, new >> sites. " >> >> And my understanding is that the Knot Atlas >> http://katlas.math.toronto.edu/wiki/Main_Page and the Mathematica >> package creating it is the state of the art. Does SnapPy now really have >> all that combinatorial stuff? I think that at the very least a good >> wrapper allowing for use of *any* robust backend for knots would be a great >> contribution to Sage. mmarco seems to have a good sense of what would >> actually be needed to do this. It's definitely a significant hole in Sage. >> - one might even wonder whether the authors of the Mma package would be >> willing to license their package in such a way that the algorithms for >> computing various invariants etc. could be used/{P,C}ythonized in Sage, if >> some people know them http://katlas.math.toronto.edu/wiki/Acknowledgement >> >> -- >> 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 <javascript:>. >> To post to this group, send email to sage-...@googlegroups.com<javascript:> >> . >> Visit this group at http://groups.google.com/group/sage-devel. >> For more options, visit https://groups.google.com/groups/opt_out. >> > > -- 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+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/groups/opt_out.
