I have gone through the package and it seems to have integrated sage and SnapPy for computing Alexander's polynomial. They have used the idea of manifolds to implement (I would like to mention that my grasp of subject is not that far even though I understand the basics of manifold as a local homeomorphism to real line( I might be completely wrong)). My plan was to start with implementing the Braid groups with the braid word, assigning numbers to the generators and reading from top to bottom which has been the most used algorithm to construct braids. Then I would like to use the concept of Braid words to get to the Alexander's polynomial (That could be achieved through Burau representation). Then other representation like Lawerence - Krammer could be achieved by relating to matrices (The points I have mentioned above have been already been implemented in various other modules). I had the idea of implementing the Kaufmann's invariant alteast for small number of crossings by the following way : As we can construct a knot from a braid, if the crossing at each point can be mentioned by X for one going over the other and X inverse for the one going below the other and then applying the conditions and splitting it for each crossing and representing the new replacements by one and the other by zero could lead to the final polynomial.I am even trying to understand the implementation of invariants like the HOMFLY - PT polynomial and Khovanov Homology (atleast the arc representation is possible to implement). My initial attempt was to relate the braids to anyon braiding which act as gates to perform quantum computation (I could not find any material regarding this but I am still on the search).My recent realization being it can be achieved as a solution to Yang Baxter Equations. These are the ideas I have had but the algorithms relating to the implementation still needs heavy thinking.
On Thu, Jan 30, 2014 at 4:23 AM, David Joyner <wdjoy...@gmail.com> wrote: > On Wed, Jan 29, 2014 at 4:32 PM, Amit <bitsjamada...@gmail.com> wrote: > > Hello, > > I would like to discuss the implementation of Braid Groups. This > > Are you planning on going beyond what is already known? > http://www.math.uiuc.edu/~nmd/snappea/ > If so, what is your plan? > > > would involve the implementation of various invariants related to Braids > > like the Alexander's polynomial > > (http://mathworld.wolfram.com/AlexanderPolynomial.html) by building up > the > > Burau representation of the same > > (http://mathworld.wolfram.com/BurauRepresentation.html) [There are more > > accurate versions of Braid Group representation] and various other > > properties relating to permutation group underlying Braids. However I > could > > not think of any idea which would implement the other invariants like the > > Kauffman's invariant for knots (I wonder whether such kind of > implementation > > can be worked around atleast for knots with less number of crossings). I > was > > also looking through the implementation of Braid Diagrams by various > means > > one attempt was by using TikZ. Braid Diagrams can be converted into link > > diagrams as every link can be represented as closed Braid. The main > > motivation behind everything is to implement certain features in Knot > Theory > > module of Mathematica > > ( > http://katlas.math.toronto.edu/wiki/The_Mathematica_Package_KnotTheory%60) > > in Sympy. Thanks. > > > > -- > > You received this message because you are subscribed to the Google Groups > > "sympy" group. > > To unsubscribe from this group and stop receiving emails from it, send an > > email to sympy+unsubscr...@googlegroups.com. > > To post to this group, send email to sympy@googlegroups.com. > > Visit this group at http://groups.google.com/group/sympy. > > For more options, visit https://groups.google.com/groups/opt_out. > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sympy+unsubscr...@googlegroups.com. > To post to this group, send email to sympy@googlegroups.com. > Visit this group at http://groups.google.com/group/sympy. > For more options, visit https://groups.google.com/groups/opt_out. > -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to sympy+unsubscr...@googlegroups.com. To post to this group, send email to sympy@googlegroups.com. Visit this group at http://groups.google.com/group/sympy. For more options, visit https://groups.google.com/groups/opt_out.
