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

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

     Presentation: MonStgElt             Default: "Artin"

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.