How to check that the Braid group has the known presentation?
Why we are sure that the presentation for the braid group is
$$
mathcal{B}_n= leftlangle sigma_1 , dots, sigma_{n-1} vert
mathcal{R}_n
rightrangle,
$$
where $$mathcal{R}_n=left{
begin{smallmatrix}
sigma_isigma_{i+1}sigma_i=sigma_{i+1}sigma_isigma_{i+1} forall 1leq ileq n-2, \ sigma_isigma_j = sigma_jsigma_i mbox{if } vert i-jvert ge 2
end{smallmatrix}
right}$$
How can we prove it? It's obvious that those relations are true in the Braid Group, but how we know that those are the unique relations needed?
The definition I'm basing is the group of isotopy classes of braids of $n$ strands.
group-theory isometry group-presentation combinatorial-group-theory braid-groups
edited Nov 30 '18 at 2:56
Shaun
8,820113681
asked Jun 14 '17 at 3:53
MonsieurGaloisMonsieurGalois
3,4231333
add a comment |
Why we are sure that the presentation for the braid group is
$$
mathcal{B}_n= leftlangle sigma_1 , dots, sigma_{n-1} vert
mathcal{R}_n
rightrangle,
$$
where $$mathcal{R}_n=left{
begin{smallmatrix}
sigma_isigma_{i+1}sigma_i=sigma_{i+1}sigma_isigma_{i+1} forall 1leq ileq n-2, \ sigma_isigma_j = sigma_jsigma_i mbox{if } vert i-jvert ge 2
end{smallmatrix}
right}$$
How can we prove it? It's obvious that those relations are true in the Braid Group, but how we know that those are the unique relations needed?
The definition I'm basing is the group of isotopy classes of braids of $n$ strands.
group-theory isometry group-presentation combinatorial-group-theory braid-groups
edited Nov 30 '18 at 2:56
Shaun
8,820113681
asked Jun 14 '17 at 3:53
MonsieurGaloisMonsieurGalois
3,4231333
1
What is your definition of the braid group?
– Eric Wofsey
Jun 14 '17 at 3:56
I understand it as the group of isotopy classes of $n$ paths that start and end in the same set of points.
– MonsieurGalois
Jun 14 '17 at 4:13
3
This is a theorem of Artin: jstor.org/stable/1969218?origin=crossref.
– Erick Wong
Jun 14 '17 at 4:31
1
A proof should be also in Joan Birman's book on the braid group.
– Moishe Cohen
Jun 14 '17 at 13:11
add a comment |
Why we are sure that the presentation for the braid group is
$$
mathcal{B}_n= leftlangle sigma_1 , dots, sigma_{n-1} vert
mathcal{R}_n
rightrangle,
$$
where $$mathcal{R}_n=left{
begin{smallmatrix}
sigma_isigma_{i+1}sigma_i=sigma_{i+1}sigma_isigma_{i+1} forall 1leq ileq n-2, \ sigma_isigma_j = sigma_jsigma_i mbox{if } vert i-jvert ge 2
end{smallmatrix}
right}$$
How can we prove it? It's obvious that those relations are true in the Braid Group, but how we know that those are the unique relations needed?
The definition I'm basing is the group of isotopy classes of braids of $n$ strands.
group-theory isometry group-presentation combinatorial-group-theory braid-groups
edited Nov 30 '18 at 2:56
Shaun
8,820113681
asked Jun 14 '17 at 3:53
MonsieurGaloisMonsieurGalois
3,4231333
Why we are sure that the presentation for the braid group is
$$
mathcal{B}_n= leftlangle sigma_1 , dots, sigma_{n-1} vert
mathcal{R}_n
rightrangle,
$$
where $$mathcal{R}_n=left{
begin{smallmatrix}
sigma_isigma_{i+1}sigma_i=sigma_{i+1}sigma_isigma_{i+1} forall 1leq ileq n-2, \ sigma_isigma_j = sigma_jsigma_i mbox{if } vert i-jvert ge 2
end{smallmatrix}
right}$$
How can we prove it? It's obvious that those relations are true in the Braid Group, but how we know that those are the unique relations needed?
The definition I'm basing is the group of isotopy classes of braids of $n$ strands.
group-theory isometry group-presentation combinatorial-group-theory braid-groups
group-theory isometry group-presentation combinatorial-group-theory braid-groups
edited Nov 30 '18 at 2:56
Shaun
8,820113681
asked Jun 14 '17 at 3:53
MonsieurGaloisMonsieurGalois
3,4231333
edited Nov 30 '18 at 2:56
Shaun
8,820113681
asked Jun 14 '17 at 3:53
MonsieurGaloisMonsieurGalois
3,4231333
edited Nov 30 '18 at 2:56
Shaun
8,820113681
edited Nov 30 '18 at 2:56
Shaun
8,820113681
edited Nov 30 '18 at 2:56
Shaun
8,820113681
8,820113681
asked Jun 14 '17 at 3:53
MonsieurGaloisMonsieurGalois
3,4231333
asked Jun 14 '17 at 3:53
MonsieurGaloisMonsieurGalois
3,4231333
asked Jun 14 '17 at 3:53
MonsieurGaloisMonsieurGalois
3,4231333
3,4231333
1
What is your definition of the braid group?
– Eric Wofsey
Jun 14 '17 at 3:56
I understand it as the group of isotopy classes of $n$ paths that start and end in the same set of points.
– MonsieurGalois
Jun 14 '17 at 4:13
3
This is a theorem of Artin: jstor.org/stable/1969218?origin=crossref.
– Erick Wong
Jun 14 '17 at 4:31
1
A proof should be also in Joan Birman's book on the braid group.
– Moishe Cohen
Jun 14 '17 at 13:11
add a comment |
1
What is your definition of the braid group?
– Eric Wofsey
Jun 14 '17 at 3:56
I understand it as the group of isotopy classes of $n$ paths that start and end in the same set of points.
– MonsieurGalois
Jun 14 '17 at 4:13
3
This is a theorem of Artin: jstor.org/stable/1969218?origin=crossref.
– Erick Wong
Jun 14 '17 at 4:31
1
A proof should be also in Joan Birman's book on the braid group.
– Moishe Cohen
Jun 14 '17 at 13:11
1
1
What is your definition of the braid group?
– Eric Wofsey
Jun 14 '17 at 3:56
What is your definition of the braid group?
– Eric Wofsey
Jun 14 '17 at 3:56
I understand it as the group of isotopy classes of $n$ paths that start and end in the same set of points.
– MonsieurGalois
Jun 14 '17 at 4:13
I understand it as the group of isotopy classes of $n$ paths that start and end in the same set of points.
– MonsieurGalois
Jun 14 '17 at 4:13
3
3
This is a theorem of Artin: jstor.org/stable/1969218?origin=crossref.
– Erick Wong
Jun 14 '17 at 4:31
This is a theorem of Artin: jstor.org/stable/1969218?origin=crossref.
– Erick Wong
Jun 14 '17 at 4:31
1
1
A proof should be also in Joan Birman's book on the braid group.
– Moishe Cohen
Jun 14 '17 at 13:11
A proof should be also in Joan Birman's book on the braid group.
– Moishe Cohen
Jun 14 '17 at 13:11
add a comment |
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1
What is your definition of the braid group?
– Eric Wofsey
Jun 14 '17 at 3:56
I understand it as the group of isotopy classes of $n$ paths that start and end in the same set of points.
– MonsieurGalois
Jun 14 '17 at 4:13
3
This is a theorem of Artin: jstor.org/stable/1969218?origin=crossref.
– Erick Wong
Jun 14 '17 at 4:31
1
A proof should be also in Joan Birman's book on the braid group.
– Moishe Cohen
Jun 14 '17 at 13:11