This week we have made an attempt at implementing the Jones Polynomial. We have used the trip matrix of the knot to determine the Jones polynomial. The trip matrix of a knot is determined by the following process. We number the crossings randomly, and we start moving along the knot, let T be the matrix and T ij be the elements. Now we start with the crossing i and see how many times we have encountered the crossing j until we return to the crossing i. We take mod2 of this value and fill that matrix element. So in this way we construct all the elements except the diagonal elements. For the diagonal elements we see whether i is a positive cross or negative cross. If it is a positive cross we fill it with zero and for the negative cross we fill it with 1. Now we have the initial trip matrix. To evaluate the Jones polynomial we smooth the crossings until we have a link for which we know the Kauffman’s bracket. So this decomposition here is looked by the matrix. So for the initial diagonal elements of the trip matrix we assign a certain type of smoothing and determine the number of seifert circles. Now we construct a new matrix by doing the following, we choose a crossing and smooth it in another way(different from the first), the only elements which are different from the initial matrix are the diagonal elements and the only element which changes when we do such kind of a smoothing is the crossing number element. In sense if we change the smoothing at crossing i we change the number at the matrix element T ii (this is flipped from either one to zero or zero to one). Again we continue this until all the options are exhausted. Then for every matrix we have certain coefficients of the jones polynomial. Adding all these up gives the jones polynomial for the knot. I know it is tough to follow but that is the gist of the algorithm. I have followed the following material and I request the readers to have a look at them for greater understanding.
Second Reference :
A matrix for computing the Jones Polynomial of a Knot by Louis Zulli
We have dedicated some time for documenting the code that we coded till now. There have been some edge cases where the code showed some inconsistency. We are working on the edge cases as well as cleaning the code alongside continuing the implementation of the invariants.
Here is the pull request for the week: