GSoC 2014 - Implementation of Braid Groups and extensions to Knots

[Amit]

Hello,
       I would like to discuss the implementation of Braid Groups. This 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. 

[David Joyner]

On Wed, Jan 29, 2014 at 4:32 PM, Amit <bitsja...@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?

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[Amit Jamadagni]

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.  

[Amit Jamadagni]

Can this be a feasible idea to be worked upon for SoC ?? Any comments would be great.