Hello,
I have started working on my proposal and I would like to present my
ideas of implementation. With my understanding I see two phases to the
project one which plays around with various representations and conversion
between them and then a separate class of invariants which would form the
second phase(But both would be worked on simultaneously, the phase split is
only to make two as independent as possible). I would like to start of with
the representation part as there seems to be work done with respect to
braid groups. We start with the braid word as the input for the knot and
then calculate the Seifert Matrix and then Alexander's polynomial from this
matrix. So once an invariant is calculated we can move to the others from
this (For reference we can use
http://mathworld.wolfram.com/AlexanderPolynomial.html. A list of total
implementation of presentation as well as invariants is given in
http://www.indiana.edu/~knotinfo/ out of which we can try to achieve as
much as possible taking into consideration the feasibility. Then once we
are done with the braid word representation I would like to implement the
Vogel's algorithm which takes in the Gauss Code and generates the braid
word. So this would be a layer above the initial layer as the user can
input either the Gauss Code or Briad word and similarly this would linked
to the Invariants class. Gauss code being closely related to DT code we can
use the above implementation to generate the DT Code. The presentations
which remain are the Planar Diagram presentation, Arc presentations, Conway
notation (This is in comparison to
http://katlas.org/wiki/The_Mathematica_Package_KnotTheory%60) for which I
have to find a way in order to convert between.Finally there is the fox
algorithm from which we can move to the Alexander's polynomial. I am yet to
see the partial differential implementation in Sage which might be useful
for this implementation. This would come under the presentation part. So to
summarize it,
class Various
presentations ============== class of invariants
|
|
inter conversion
between
one presentation to
other ---------------------------- >same here
|
|
More generally
this would
help in taking the
input for ========> This could be extended to present the various diagrams.
various
computations
* I feel taking in the input would be the most difficult part even though
we would provide with a lot of options, I guess user would be more
interested in giving in a input which is as compact as possible.
I understand the need to focus more on the invariants and I will try to add
(mainly the algorithms) as much as possible in the coming days.
Finally to test the above implementation we can use the already present
software which I would like to list in order to what goes for what.
1. For the Siefert Matrix we have the site
http://www.maths.ed.ac.uk/~s0681349/SeifertMatrix/#alphabetical
2. For Vogel's algorithm we have the GAP implementation ( I still have to
go through this).
3. For braid to DT code we can use knotscape to test the code.
I hope the above outline would take around 4 - 5 weeks with everything in
place (Considering the documentation, test cases, code including a part of
the invariants). The rest of the remaining weeks if everything goes
according to plan would be implementing the other invariants.
I have seen through Spherogram which has a very good implementation of
links. I am still in the process of reading the entire material and would
integrate parts of it once I am thoroughly done with it.
This is an outline on which I would like to build my proposal. Any comments
or further inputs would be of great help in making this project successful.
Amit.
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