Constructing and Accessing Braid Groups

This section describes the facilities for creating a braid group and for accessing and changing its basic properties.

BraidGroup(n: parameters): RngIntElt -> GrpBrd

BraidGroup(GrpBrd, n: parameters): Cat, RngIntElt -> GrpBrd

Presentation: MonStgElt                    Default: "Artin"

Given a small integer \(n>0\), return the braid group on \(n\) strings, that is, \(n-1\) Artin generators.

The presentation can be selected using the parameter Presentation. Possible values for this parameter are "Artin" (default) and "BKL".

GetPresentation(B): GrpBrd -> MonStgElt

Returns a string \(s\), either "Artin" or "BKL", indicating the presentation currently used for \(B\).

SetPresentation(~B, s): GrpBrd, MonStgElt

Set the presentation used for \(B\) to the presentation indicated by \(s\). Possible values for \(s\) are "Artin" and "BKL".

GetForceCFP(B): GrpBrd -> BoolElt

Returns whether arithmetic operations with elements of \(B\) are always done using representations of the elements as products of simple elements.

SetForceCFP(~B, b): GrpBrd, BoolElt

By default, arithmetic operations with elements of \(B\) are always done using representations of the elements as products of simple elements. If necessary, such representations are computed. Experienced users can turn this feature off for a braid group \(B\) using the procedure SetForceCFP with \(b\) set to false.

GetElementPrintFormat(B): GrpBrd -> MonStgElt

Returns a string \(s\) indicating the format currently used for printing elements of \(B\). \(s\) is one of "Word", "CFP" or "Both".

SetElementPrintFormat(~B, s): GrpBrd, MonStgElt

When printing an element of a braid group \(B\), by default both the representation as word in the generators and the representation as product of simple elements in the presentation selected for \(B\) are printed. Experienced users can use the procedure SetElementPrintFormat for changing the print format. Possible values for \(s\) are "Word", "CFP" and "Both".

NumberOfStrings(B): GrpBrd -> RngIntElt

Given a braid group \(B\), return the number of strings on which \(B\) is defined.

NumberOfGenerators(B): GrpBrd -> RngIntElt

Ngens(B): GrpBrd -> RngIntElt

Given a braid group \(B\), return the number of Artin generators of \(B\). Note that the number of Artin generators is returned regardless of the presentation selected for \(B\).