Some improvements for braids computations

[enriqueartal]

Is there an existing issue for this?

• I have searched the existing issues for a bug report that matches the one I want to file, without success.

Problem Description

• The method conjugating_braid produces, if any, a conjugating braid longer than expected.
• The former method left_normal_form was replaced to be included in the more general method with the same name for Artin groups. The former method was much faster for braid groups.
• When two braids are conjugated by a pure braid, it may be interesting to get such pure braid.

Proposed Solution

• For left_normal_form, mimic the right_normal_form method with the suitable changes. Both left_normal_form methods produce the same result but after the change it is faster.
• For conjugating_braid, the former method gave a braid in normal form and even powers of Delta can be avoided since it is central.
• Creat a new method pure_conjugating_braid; it produces None if the braids are either non conjugate or the associated permutations are different. If not, conjugating_braid is applied and then we multiply it by a braid centralizing self whose permutation is inverse to the conjugating braid.

Alternatives Considered

See in Proposed Solution.

Additional Information

The solution is written in the branch braids of the fork https://github.com/enriqueartal/sage. A pull request has been added.