Some improvements for braids computations #35214

enriqueartal · 2023-02-27T13:06:55Z

📚 Description

Fasten the computation of the left normal form of a braid, produce a shorter conjugating braid and if possible a pure conjugating braid

The method left_normal_form for braids was modified to be part of the general method for Artin groups. I propose to come back to the former method, it produces the same result but much faster. I added left_normal_form as a method for braid group and kept _left_normal_form_coxeter though it is not used any more for braids.
The method conjugating_braid produces a braid in left normal form with high powers of Delta, which can be reduce to 0 or 1.
If two braids are conjugated and have the same permutation, the conjugation can be realized by a pure braid. Starting from conjugating_braid a simple computation produces such a pure braid.

There are mild improvements for the treatment of braid groups.

Resolves #35213

📝 Checklist

I have made sure that the title is self-explanatory and the description concisely explains the PR.
I have linked an issue or discussion.
I have created tests covering the changes.
I have updated the documentation accordingly.

⌛ Dependencies

trevorkarn · 2023-02-27T20:19:57Z

src/sage/groups/braid.py

-        """
-        Return the right normal form of the braid.
+        r"""
+        A tuple of simple generators in the right normal form. The last


You could leave the original "Return the right normal form" and have your additions on the next line after a blank line.

trevorkarn · 2023-02-27T20:22:31Z

src/sage/groups/braid.py

@@ -1620,21 +1644,31 @@ def conjugating_braid(self, other):

            sage: B = BraidGroup(3)
            sage: a = B([2, 2, -1, -1])
-            sage: b = B([2, 1, 2, 1])
+            sage: b = B([2, 2, 2, 2, 1])


Maybe I'm not understanding something, but you could leave the already existing tests and just add to them below.

trevorkarn · 2023-02-27T20:25:33Z

src/sage/groups/braid.py

+            sage: c = B([1])
+            sage: c.left_normal_form()
+            (1, s0)
+        """


You might consider adding tests for trivial examples.

Thanks for your comments. I completed the descriptions and added more examples.

codecov-commenter · 2023-02-28T01:38:24Z

Codecov Report

Patch coverage: 95.45% and project coverage change: -0.02 ⚠️

Comparison is base (c00e6c2) 88.62% compared to head (c7e4d83) 88.60%.

Additional details and impacted files

@@             Coverage Diff             @@
##           develop   #35214      +/-   ##
===========================================
- Coverage    88.62%   88.60%   -0.02%     
===========================================
  Files         2148     2148              
  Lines       398855   398894      +39     
===========================================
- Hits        353480   353452      -28     
- Misses       45375    45442      +67

Impacted Files	Coverage Δ
src/sage/groups/braid.py	`96.79% <95.12%> (-0.09%)`	⬇️
src/sage/groups/artin.py	`95.39% <100.00%> (+0.03%)`	⬆️

... and 23 files with indirect coverage changes

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enriqueartal · 2023-02-28T10:10:47Z

Fixes #35213

trevorkarn · 2023-02-28T21:22:51Z

It would be good to have full coverage. Is it easy to see which of your additions don't have a test associated to them?

enriqueartal · 2023-02-28T22:33:25Z

For left_normal_form, I copied the examples of the former method (in artin.py) and add some more examples. I could add something showing that now it is much faster if needed.
For conjugating_braid it is a mild change from the original method, I keep now the examples and I add some more to see how the conjugating braid is shorter. I have added in the last commit 56d33d8 an example of non conjugation.
For the new method pure_conjugating_braid, I have added in the same commit a couple of examples where the output is None

… the left normal form was a power of Delta to ensure all the factors of a product are braids - In pure_conjugating_braid, check if the permutation of the conjugating braid is in the subgroup generated by the permutations of the centralizing braids of self; return None if not.

enriqueartal · 2023-03-02T08:34:13Z

Commit 690c5d5 solves a couple of problems. When the conjugating braid was trivial, as coded I multiplied a braid and an integer yielding an error. For pure_conjugating_braid, cases where a conjugating braid exists but not a pure conjugating braid were not considered and the output was an error instead of None.

trevorkarn · 2023-03-09T01:46:52Z

I won't be able to get to this until next week. If I haven't given any comments by the end of next week, please ping me again :)

tscrim

You should not hide other computational methods but expose both through an, e.g., algorithm argument. This is not just good for testing purposes (which you should do a more systematic comparison between the two methods), but also gives additional options to the user.

tscrim · 2023-03-13T09:13:24Z

src/sage/groups/braid.py

+        if p3 not in G.subgroup(LP):
+            return None
+        P = p3.word_problem(LP, display = False, as_list = True)
+        b1 = prod(LB[LP.index(G(a))] ** b for a,b in P)


What happens if P is empty? Add a doctest checking for this corner case (if it can occur).

Additionally, if you make LP a dict from permutations to braids, this becomes faster and you can still pass it off to G.subgroup().

For the first question a previous if makes impossible to have P empty. I could not pass the dict to subgroup and even if it would be possible, I think I cannot pass it to word_problem

For the first question a previous if makes impossible to have P empty.

I might say also the previous p3.is_one() also makes that the case. However, again this case should have a doctest.

I could not pass the dict to subgroup and even if it would be possible, I think I cannot pass it to word_problem

You can pass a dict to subgroup:

sage: G = groups.permutation.Symmetric(5) sage: d = {G.random_element(): i for i in range(5)} sage: d {(1,5,2,4): 0, (1,5)(2,3): 1, (1,3,5,2): 2, (1,5,3,2): 3, (1,5)(2,4,3): 4} sage: G.subgroup(d) Subgroup generated by [(1,3,5,2), (1,5,3,2), (1,5)(2,3), (1,5)(2,4,3), (1,5,2,4)] of (Symmetric group of order 5! as a permutation group)

Indeed for word_problem, you cannot pass the dict, but you can do list(d), which will have a list in the same order the dict d was constructed (Python3 guarantees this).

src/sage/groups/braid.py

tscrim · 2023-03-13T09:17:32Z

src/sage/groups/braid.py

+    def pure_conjugating_braid(self, other):
+        r"""
+        Return a pure conjugating braid, if it exists.
+


Add a more detailed description; in particular, including the definition of a pure conjugating braid.

It should not be part of the one-line description. Spend a little bit of time/mathematical notation spelling it out a bit more.

tscrim · 2023-03-13T09:17:48Z

src/sage/groups/braid.py

+
+        INPUT:
+
+        - ``other`` -- the other braid to look for conjugating braid


This is confusing.

tscrim · 2023-03-13T09:19:05Z

src/sage/groups/braid.py

+            n = B.strands()
+            k = int(l[0][0] % 2)
+            b0 = B.delta() ** k * B(prod(B(a) for a in l[1:]))
+            return b0


Suggested change

n = B.strands()

k = int(l[0][0] % 2)

b0 = B.delta() ** k * B(prod(B(a) for a in l[1:]))

return b0

l[0][0] %= 2

return B._element_from_libbraiding(l)

tscrim · 2023-03-13T09:19:39Z

src/sage/groups/braid.py

+            sage: b = B([2, 2, 2, 2, 1])
+            sage: c = b * a / b
+            sage: d1 = a.conjugating_braid(c)
+            sage: print (len(d1.Tietze()))


PEP8

Suggested change

sage: print (len(d1.Tietze()))

sage: print(len(d1.Tietze()))

and similar below.

tscrim · 2023-03-13T09:20:16Z

src/sage/groups/braid.py

+            sage: a1=B([1])
+            sage: a2=B([2])


PEP8

Suggested change

sage: a1=B([1])

sage: a2=B([2])

sage: a1 = B([1])

sage: a2 = B([2])

tscrim · 2023-03-13T09:20:58Z

src/sage/groups/braid.py

+        B = self.parent()
+        n = B.strands()
+        G = SymmetricGroup(n)
+        p1 = G(self.permutation())


I would only convert to G when it becomes necessary.

I move the definition of G few lines below, and eliminate some conversions

tscrim · 2023-03-13T09:21:45Z

src/sage/groups/braid.py

+        if p1 != p2:
+            return None
+        b0 = self.conjugating_braid(other)
+        if not b0:


Suggested change

if not b0:

if b0 is not None:

It is better to check explicitly against None.

Done but putting if b0 is None

Ah, right, whoops. Thanks.

Simplify code for conjugating_braid Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

enriqueartal · 2023-03-20T15:43:18Z

Done, with longer explanations for the commit.

tscrim

Thank you. LGTM.

…l, TL_representation and links_gould_polynomial

enriqueartal · 2023-03-25T23:11:14Z

I found some small typos in the documentation, @trevorkarn, could you take a look?

tscrim · 2023-03-26T00:07:39Z

@enriqueartal You need to take care of the invalid escape sequence by making the start of the doc string r”””; see the lint report.

github-actions · 2023-03-26T10:10:05Z

Documentation preview for this PR is ready! 🎉
Built with commit: c7e4d83

tscrim · 2023-03-28T02:12:45Z

src/sage/groups/braid.py

+        A conjugating braid.
+
+        More precisely, if the output is `d`, `o` equals ``other``, and `s` equals ``self``
+        then `o = d^{-1} \cdot s \cdot d`.


One last trivial thing: This needs a blankline after it.

Done, and also style spaces all around

tscrim

A lot of those were not PEP8 violations (higher precedence operators do not need the same spacing, which include [] and ()), and some of them actually makes the code harder to read. However, the rule was not applied consistently. I would actually just revert those additional PEP8 spacing changes as it makes the patch larger for minimal (at best) gain. Better separation of ticket purpose.

enriqueartal · 2023-03-29T06:26:31Z

Done. I found curious that rules were not applied consistently even in the same line, but I just kept the required line. Unless some changes are asked by you or by some other person, I stop working in this PR.

tscrim

Thank you. I think this is ready to go in!

enriqueartal · 2023-03-29T08:05:33Z

@tscrim thanks for your help!

…ations, and a 2-generator presentation  This PR is a continuation of sagemath#35214. - Given an element of a symmetric group, the associated permutation braid can be constructed, added in `_element_constructor`. - The method `_standard_lift_Tietze` has been simplified (for a permutation) and now it gives a shortest word for the lift but not necesarily the smallest for the lexicographic order. The former code has been commented. - The method `presentation2gens` provides a presentation of the braid group with two generators. The general method `epimorphism` has been adapted to braid groups using this presentation and it is much faster. ### 📝 Checklist     - [x] The title is concise, informative, and self-explanatory. - [x] The description explains in detail what this PR is about. - [ ] I have linked a relevant issue or discussion. - [x] I have created tests covering the changes. - [x] I have updated the documentation accordingly. ### ⌛ Dependencies   URL: sagemath#35985 Reported by: Enrique Manuel Artal Bartolo Reviewer(s): Enrique Manuel Artal Bartolo, Travis Scrimshaw

enriqueartal added 5 commits February 26, 2023 14:25

conjugating

670fc7a

first version of a shorter conjugating_braid

74ffbd3

adding pure_conjugating_braid method

362108a

some trail white spaces

960e0b4

add left_normal_form and simplify previous code

5f1952b

trevorkarn reviewed Feb 27, 2023

View reviewed changes

trevorkarn added s: needs review c: group theory and removed s: needs review labels Feb 27, 2023

enriqueartal added 2 commits February 27, 2023 22:55

Description and more examples

c158a0a

extra spaces removed

5c42b98

Move a comment from TODO to NOTE

eea3f55

enriqueartal closed this Feb 28, 2023

enriqueartal reopened this Feb 28, 2023

More examples

56d33d8

Merge branch 'sagemath:develop' into braids

89d50fb

enriqueartal requested a review from trevorkarn March 8, 2023 07:39

Merge branch 'sagemath:develop' into braids

eef2f8d

tscrim requested changes Mar 13, 2023

View reviewed changes

Update src/sage/groups/braid.py

fedd35e

Simplify code for conjugating_braid Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

tscrim approved these changes Mar 21, 2023

View reviewed changes

tscrim added p: major / 3 s: positive review t: enhancement and removed s: needs review labels Mar 21, 2023

There were typos in the docs of reduced_burau_matrix, jones_polynomia…

d92e0e5

…l, TL_representation and links_gould_polynomial

tscrim approved these changes Mar 26, 2023

View reviewed changes

tscrim self-requested a review March 26, 2023 00:07

lint problems

c7e4d83

Merge branch 'sagemath:develop' into braids

1cd0e79

tscrim reviewed Mar 28, 2023

View reviewed changes

enriqueartal added 2 commits March 28, 2023 08:59

Added blank line and plenty of style spaces

39ce62e

Some lint typos

57f49f1

tscrim requested changes Mar 29, 2023

View reviewed changes

enriqueartal added 3 commits March 29, 2023 08:16

Revert style changes, except blank line

791c9ae

Merge branch 'braids' of github.com:enriqueartal/sage into braids

6041d12

Keep only required blank line

30d6be1

tscrim approved these changes Mar 29, 2023

View reviewed changes

vbraun merged commit 42817c2 into sagemath:develop Apr 1, 2023

mkoeppe added this to the sage-10.0 milestone Apr 1, 2023

mkoeppe removed the s: positive review label Apr 1, 2023

enriqueartal deleted the braids branch April 8, 2023 09:52

enriqueartal mentioned this pull request Jul 24, 2023

More improvements for braid groups involving permutations, and a 2-generator presentation #35985

Merged

5 tasks


		INPUT:

		- ``other`` -- the other braid to look for conjugating braid

Some improvements for braids computations #35214

Some improvements for braids computations #35214

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enriqueartal commented Feb 27, 2023 • edited

📚 Description

📝 Checklist

⌛ Dependencies

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