# Difference between revisions of "ApCoCoA-1:Braid groups"

From ApCoCoAWiki

Line 3: | Line 3: | ||

The Braid groups are infinite for a natural number n > 1 and have the following presentation. | The Braid groups are infinite for a natural number n > 1 and have the following presentation. | ||

B(n) = <g_{1},...,g_{n-1} | g_{i}g_{j} = g_{j}g_{i} for |i-j| >= 2, g_{i}g_{i+1}g_{i} = g_{i+1}g_{i}g_{i+1} for 1 <= i <= n-2> | B(n) = <g_{1},...,g_{n-1} | g_{i}g_{j} = g_{j}g_{i} for |i-j| >= 2, g_{i}g_{i+1}g_{i} = g_{i+1}g_{i}g_{i+1} for 1 <= i <= n-2> | ||

− | The complexity in the group B(n) grows with n. We get the trivial group for n = 1 and the infinite cyclic group for n = 2. | + | The complexity in the group B(n) grows with n. We get the trivial group for n = 1 and the infinite cyclic group for n >= 2. |

(Reference: E. Artin, "Theory of braids" Ann. of Math. , 48 (1947) pp. 643–649 and | (Reference: E. Artin, "Theory of braids" Ann. of Math. , 48 (1947) pp. 643–649 and |

## Revision as of 07:08, 23 August 2013

#### Description

The Braid groups are infinite for a natural number n > 1 and have the following presentation.

B(n) = <g_{1},...,g_{n-1} | g_{i}g_{j} = g_{j}g_{i} for |i-j| >= 2, g_{i}g_{i+1}g_{i} = g_{i+1}g_{i}g_{i+1} for 1 <= i <= n-2>

The complexity in the group B(n) grows with n. We get the trivial group for n = 1 and the infinite cyclic group for n >= 2.

(Reference: E. Artin, "Theory of braids" Ann. of Math. , 48 (1947) pp. 643–649 and

W. Magnus, Braid groups: A survey, Proceedings of the Second International Conference on the Theory of Groups, Canberra, Australia, 1973, pp. 463-487.)

#### Computation

We enumerate partial Groebner bases for the Braid groups as follows.

/*Use the ApCoCoA package ncpoly.*/ // Number of Braid group MEMORY.N:=3; Use ZZ/(2)[a[1..MEMORY.N],b[1..MEMORY.N]]; NC.SetOrdering("LLEX"); Define CreateRelationsBraid() Relations:=[]; For Index1:= 1 To MEMORY.N Do Append(Relations,[[a[Index1],b[Index1]],[1]]); Append(Relations,[[b[Index1],a[Index1]],[1]]); EndFor; For Index2:=1 To MEMORY.N Do For Index3:=(Index2+2) To MEMORY.N Do If Abs(Index2-Index3)>1 Then // Insert the relation a_{i}a_{i+2} = a_{i+2}a_{i} Append(Relations,[[a[Index2],a[Index3]],[a[Index3],a[Index2]]]); EndIf; EndFor; EndFor; For Index4:=1 To MEMORY.N Do For Index5:=(Index4+1) To MEMORY.N Do If Abs(Index4-Index5)=1 Then // Insert the relation a_{i}a_{i+1}a_{i} = a_{i+1}a_{i}a_{i+1} Append(Relations,[[a[Index4],a[Index5],a[Index4]],[a[Index5],a[Index4],a[Index5]]]); EndIf EndFor; EndFor; Return Relations; EndDefine; Relations:=CreateRelationsBraid(); -- Enumerate a partial Groebner basis (see NC.GB for more details) NC.GB(G,31,1,100,1000);