Hello everyone,

The past week has been really on the thinking end. The task was to detect the braid word from the oriented gauss code. So I mentioned the dots were to be connected, but then I had problems implementing it. I was thinking more and more rather than a looking for the straight implementation. It took me a week to break through and then I had some consistent results. So they were various ideas on how one could detect the braidword. One was to update the outgoing strands after each crossing has been looked into and then match it with were we have the incoming. I was working on various other ideas as well as I felt there was some kind of difficulty on the implementation end. Here are the gists of code on the ideas I worked upon.

The ones I attempted

https://gist.github.com/amitjamadagni/1ccad5ca91b96eac367b

The one which is working

https://gist.github.com/amitjamadagni/57b5de098efe11056efa

This was like I had all the answers yet I was missing something. There are some glitches still though, small condition on the seifert circles one is missing. I need to work on the documentation end as we are missing on that. And here is the pull request for the past two weeks.

https://github.com/miguelmarco/sage/pull/9

Lot of work as to be done on the documenting and testing side of the code. Here are the results for the algorithm

sage: from sage.knots import link

sage: L = link.Link(oriented_gauss_code = [[-1, +2, -3, 4, +5, +1, -2, +6, +7, 3, -4, -7, -6,-5],[‘-‘,’-‘,’-‘,’-‘,’+’,’-‘,’+’]])

Seifert Circles

sage: L.seifert_circles()

[[9, 13, 9], [4, 12, 10, 4], [2, 8, 14, 6, 2], [1, 7, 3, 11, 5, 1]]

PD Code

sage: L.PD_code()

[[1, 7, 2, 6],

[7, 3, 8, 2],

[3, 11, 4, 10],

[11, 5, 12, 4],

[14, 5, 1, 6],

[13, 9, 14, 8],

[12, 9, 13, 10]]

Regions

sage: L.regions()

[[4, -11],

[2, -7],

[6, -1],

[13, 9],

[-4, -10, -12],

[-8, -2, -6, -14],

[10, -3, 8, -13],

[14, -5, 12, -9],

[7, 3, 11, 5, 1]]

We perform a move if they are bad regions

sage: L.remove_regions()

[[1, 9, 2, 8],

[9, 3, 10, 2],

[5, 13, 6, 12],

[13, 7, 14, 6],

[18, 7, 1, 8],

[17, 11, 18, 10],

[14, 11, 15, 12],

[16, 4, 17, 3],

[15, 4, 16, 5]]

Final method gives the above information after all the moves are made

sage: L.final()

Seifert Circles

[[[4, 18, 4],

[15, 21, 15],

[1, 9, 3, 19, 11, 17, 5, 13, 7, 1],

[2, 10, 20, 16, 12, 6, 14, 22, 8, 2]],

Regions

[[-15, -21],

[6, -13],

[2, -9],

[8, -1],

[18, 4],

[-3, 10, -19],

[12, -5, -17],

[20, 16, -11],

[14, 22, -7],

[19, 11, 17, -4],

[21, -14, -6, -12, -16],

[15, -20, -10, -2, -8, -22],

[5, 13, 7, 1, 9, 3, -18]],

PD Code

[[1, 9, 2, 8],

[9, 3, 10, 2],

[5, 13, 6, 12],

[13, 7, 14, 6],

[22, 7, 1, 8],

[19, 11, 20, 10],

[16, 11, 17, 12],

[18, 4, 19, 3],

[17, 4, 18, 5],

[21, 14, 22, 15],

[20, 16, 21, 15]]]

And finally we convert it to braid

sage: L.seifert_to_braid()

[-1, -2, -3, 2, 1, -2, -2, 3, 2, -2, -2]

One more example :

sage: from sage.knots import link

sage: L = link.Link(oriented_gauss_code = [[-1, +2, 3, -4, 5, -6, 7, 8, -2, -5, +6, +1, -8, -3, -4, -7],[‘-‘,’-‘,’-‘,’-‘,’+’,’+’,’-‘,’+’]])

sage: L.seifert_circles()

[[2, 10, 6, 12, 2], [4, 16, 8, 14, 4], [1, 13, 9, 3, 15, 5, 11, 7, 1]]

sage: L.regions()

[[6, -11],

[15, -4],

[9, 3, -14],

[2, -9, -13],

[1, 13, -8],

[12, -1, -7],

[5, 11, 7, -16],

[-3, 10, -5, -15],

[-6, -10, -2, -12],

[16, 8, 14, 4]]

sage: L.PD_code()

[[1, 13, 2, 12],

[9, 3, 10, 2],

[14, 4, 15, 3],

[4, 16, 5, 15],

[10, 5, 11, 6],

[6, 11, 7, 12],

[16, 8, 1, 7],

[13, 8, 14, 9]]

sage: L.remove_regions()

‘No move required’

sage: L.final()

[[[2, 10, 6, 12, 2], [4, 16, 8, 14, 4], [1, 13, 9, 3, 15, 5, 11, 7, 1]],

[[6, -11],

[15, -4],

[9, 3, -14],

[2, -9, -13],

[1, 13, -8],

[12, -1, -7],

[5, 11, 7, -16],

[-3, 10, -5, -15],

[-6, -10, -2, -12],

[16, 8, 14, 4]],

[[1, 13, 2, 12],

[9, 3, 10, 2],

[14, 4, 15, 3],

[4, 16, 5, 15],

[10, 5, 11, 6],

[6, 11, 7, 12],

[16, 8, 1, 7],

[13, 8, 14, 9]]]

sage: L.seifert_to_braid()

[-1, 2, -1, -2, -2, 1, 1, -2]

So this week we are looking at jones polynomial implementation and lots of documentation and testing.Thanks for reading through.