Braid Groups [HB 29]

The implementation of braid groups in Magma has been revised completely for version V2.10. Working with both Artin's original presentation and the band generator presentation introduced by Birman, Ko and Lee is now fully supported and elements can be represented as words or in terms of simple elements as desired. Both presentations and different representations of elements can be used simultaneously.

New features:

• The following functions for controlling for a given braid group B the default presentation used for B, the element representation used for arithmetic operations with elements of B, and the print format for elements of B have been provided: GetPresentation, SetPresentation, GetForceCFP, SetForceCFP, GetElementPrintFormat and SetElementPrintFormat.
• For accessing representations of elements, the functions WordToSequence, Infimum, Supremum, CanonicalLength and CanonicalFactorRepresentation have been provided.
• Various normal forms of elements can be computed with the new functions LeftNormalForm, RightNormalForm, LeftMixedCanonicalForm and RightMixedCanonicalForm.
• The new functions Cycle and Decycle implement the cycling and decycling operations introduced by Garside.
• The new boolean predicates IsSimple, IsSuperSummitRepresentative and IsConjugate for elements have been introduced. Moreover, the partial orderings on the elements of a braid group can be accessed with the new operators ge and le and with the predicates IsGE and IsLE. The latter versions allow specification of a presentation.
• The functions LeftGCD, RightGCD, LeftLCM and RightLCM, providing the lattice operations for elements of a braid group, have been introduced.
• The set of positive conjugates of an element and the super summit set of an element can be computed with the new functions PositiveConjugates and SuperSummitSet, respectively. Alternatively, these sets can be computed using the interactive versions of these tools.
• The functions SuperSummitInfimum, SuperSummitSupremum and SuperSummitCanonicalLength, computing invariants of the conjugacy class of an element, have been provided.
• The symmetric representation, the integral Burau representation and the modular Burau representations of a braid group can be created using the new functions SymmetricRepresentation and BurauRepresentation.

Changes:

• Pseudo random elements of a braid group can now be created using each of the possible representations of elements. The relevant functions are Random, RandomWord and RandomCFP.
• Arithmetic operations now are by default performed using representation of the operands in terms of simple elements. In general, this yields much better performance.

Removals:

• The function NormalFormWord has been removed, as it was made obsolete by the more general representation used for elements and does not correspond to the new design of the category.