https://github.com/acssiohm/tipe_braids

https://github.com/acssiohm/tipe_braids

Last synced: 26 days ago
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• Host: GitHub
• URL: https://github.com/acssiohm/tipe_braids
• Owner: Acssiohm
• Created: 2024-12-08T15:25:12.000Z (about 1 month ago)
• Default Branch: main
• Last Pushed: 2024-12-08T15:54:09.000Z (about 1 month ago)
• Last Synced: 2024-12-08T16:31:32.130Z (about 1 month ago)
• Language: Python
• Size: 3.91 KB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md

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README

        
# Braids Homotopy 

## Create a Braid

Each generator of the Braids group is associated with its number ( strictly positive ), its inverse is associated with the oppsit of the number, and 0 represents the neutral element :

- sigma_k -> k

- sigma_k^-1 -> -k

- epsilon -> 0

Then a braid is a word on those elements, that we represent with a list :

sigma_i.sigma_j.sigma_k^-1  -->  `[i, j, -k]`

Thus to create a braid, with l its associated list you can do :

`b = Braid(l)`

## Operations on braids 

`b1 = Braid(l1)

b2 = Braid(l2)`

- Concatenation / product : `b_product = b1*b2`

- Power : `b_power_k = b**k`

    Note that the power can be negative or zero ( but must be an integer )

- Divison : `b_fraction = b1/b2`

- Comparison/Test equivalence : `is_equivalent = (b1 == b2)`

- Methods of testing homotopy :

    - "Retournement de sous-mots" : `b.retournement_sous_mots()`

    - "Réduction de poignées"(default) : `b.reduction_poignees()`

- Print :

    Example : `b = Braid([1, 2, 0, -2])`

    - Replacing with letters (default) : 

         `print(b)` or `print(b.to_letters())` prints `ab.B`

    - With numbers :

        `print(b.to_string())` prints `(1)(2)(0)(-2)`