# Braid Groups

- [Introduction](braid-groups-introduction.md)

  - [Lattice Structure and Simple Elements](braid-groups-introduction.md#lattice-structure-and-simple-elements)

  - [Representing Elements of a Braid Group](braid-groups-introduction.md#representing-elements-of-a-braid-group)

    - [Automatic Conversions](braid-groups-introduction.md#automatic-conversions)

    - [Default Presentations](braid-groups-introduction.md#default-presentations)

    - [Representation Used for Group Operations](braid-groups-introduction.md#representation-used-for-group-operations)

    - [Printing of Elements](braid-groups-introduction.md#printing-of-elements)

  - [Normal Form for Elements of a Braid Group](braid-groups-introduction.md#normal-form-for-elements-of-a-braid-group)

  - [Mixed Canonical Form and Lattice Operations](braid-groups-introduction.md#mixed-canonical-form-and-lattice-operations)

  - [Conjugacy Testing and Conjugacy Search](braid-groups-introduction.md#conjugacy-testing-and-conjugacy-search)

    - [Definition of the Class Invariants](braid-groups-introduction.md#definition-of-the-class-invariants)

    - [Computing the Class Invariants](braid-groups-introduction.md#computing-the-class-invariants)

    - [Conjugacy Testing and Conjugacy Search](braid-groups-introduction.md#braid-groups-introduction-conjugacy-conjugacy)

- [Constructing and Accessing Braid Groups](braid-groups-construction.md)

  - [`BraidGroup(n: parameters): RngIntElt → GrpBrd`](braid-groups-construction.md#function-grpbrd-braidgroup)

  - [`BraidGroup(GrpBrd, n: parameters): Cat, RngIntElt → GrpBrd`](braid-groups-construction.md#function-braidgroup-cat-rngintelt)

  - [`GetPresentation(B): GrpBrd → MonStgElt`](braid-groups-construction.md#function-grpbrd-getpresentation)

  - [`SetPresentation(~B, s): GrpBrd, MonStgElt`](braid-groups-construction.md#function-grpbrd-setpresentation)

  - [`GetForceCFP(B): GrpBrd → BoolElt`](braid-groups-construction.md#function-grpbrd-getforcecfp)

  - [`SetForceCFP(~B, b): GrpBrd, BoolElt`](braid-groups-construction.md#function-grpbrd-setforcecfp)

  - [`GetElementPrintFormat(B): GrpBrd → MonStgElt`](braid-groups-construction.md#function-grpbrd-getelementprintformat)

  - [`SetElementPrintFormat(~B, s): GrpBrd, MonStgElt`](braid-groups-construction.md#function-grpbrd-setelementprintformat)

  - [`NumberOfStrings(B): GrpBrd → RngIntElt`](braid-groups-construction.md#function-numberofstrings-grpbrd)

  - [`NumberOfGenerators(B): GrpBrd → RngIntElt`](braid-groups-construction.md#function-numberofgenerators-grpbrd)

  - [`Ngens(B): GrpBrd → RngIntElt`](braid-groups-construction.md#function-ngens-grpbrd)

- [Creating Elements of a Braid Group](braid-groups-element-creation.md)

  - [`Representative(B): GrpBrd → GrpBrdElt`](braid-groups-element-creation.md#function-representative-grpbrd)

  - [`Rep(B): GrpBrd → GrpBrdElt`](braid-groups-element-creation.md#function-rep-grpbrd)

  - [`Identity(B): GrpBrd → GrpBrdElt`](braid-groups-element-creation.md#function-identity-grpbrd)

  - [`Id(B): GrpBrd → GrpBrdElt`](braid-groups-element-creation.md#function-id-grpbrd)

  - [`B ! 1: GrpBrd, RngIntElt → GrpBrdElt`](braid-groups-element-creation.md#operation-op-grpbrd-rngintelt)

  - [`FundamentalElement(B: parameters): GrpBrd → GrpBrdElt`](braid-groups-element-creation.md#function-fundamentalelement-grpbrd)

  - [`Generators(B: parameters): GrpBrd → [ GrpBrd ]`](braid-groups-element-creation.md#function-generators-grpbrd)

  - [`B . i: GrpBrd, RngIntElt → GrpBrdElt`](braid-groups-element-creation.md#operation-operation-grpbrd-rngintelt-grpbrdelt)

  - [`B . T: GrpBrd, Tup → GrpBrdElt`](braid-groups-element-creation.md#operation-operation-grpbrd-tup-grpbrdelt)

  - [`B ! [ i₁, ..., iₖ ]: GrpBrd, [ RngIntElt ] → GrpBrdElt`](braid-groups-element-creation.md#operation-op-grpbrd-rngintelt-2)

  - [`B ! [ T₁, ..., Tₖ ]: GrpBrd, [ Tup ] → GrpBrdElt`](braid-groups-element-creation.md#operation-op-grpbrd-tup)

  - [`B ! p: GrpBrd, GrpPermElt → GrpBrdElt`](braid-groups-element-creation.md#operation-grpbrd-coerce-perm)

  - [`B ! [ p₁, ...,pₖ ]: GrpBrd, [ GrpPermElt ] → GrpBrdElt`](braid-groups-element-creation.md#operation-op-grpbrd-grppermelt)

  - [`B ! T: GrpBrd, Tup → GrpBrdElt`](braid-groups-element-creation.md#operation-grpbrd-coerce-tuple)

  - [`IsProductOfParallelDescendingCycles(p): GrpPermElt → BoolElt`](braid-groups-element-creation.md#function-isproductofparalleldescendingcycles-grppermelt)

  - [`Random(B, r, s, m, n: parameters): GrpBrd, RngIntElt, RngIntElt, RngIntElt, RngIntElt → GrpBrdElt`](braid-groups-element-creation.md#function-random-grpbrd-rngintelt-rngintelt-rngintelt-rngintelt)

  - [`RandomCFP(B, r, s, m, n: parameters): GrpBrd, RngIntElt, RngIntElt → GrpBrdElt`](braid-groups-element-creation.md#function-randomcfp-grpbrd-rngintelt-rngintelt)

  - [`Random(B: parameters): GrpBrd → GrpBrdElt`](braid-groups-element-creation.md#function-random-grpbrd)

  - [`RandomCFP(B: parameters): GrpBrd → GrpBrdElt`](braid-groups-element-creation.md#function-randomcfp-grpbrd)

  - [`Random(B, m, n: parameters): GrpBrd, RngIntElt, RngIntElt → GrpBrdElt`](braid-groups-element-creation.md#function-random-grpbrd-rngintelt-rngintelt)

  - [`RandomWord(B, m, n: parameters): GrpBrd, RngIntElt, RngIntElt → GrpBrdElt`](braid-groups-element-creation.md#function-randomword-grpbrd-rngintelt-rngintelt)

  - [`RandomWord(B: parameters): GrpBrd → GrpBrdElt`](braid-groups-element-creation.md#function-randomword-grpbrd)

  - [`Example: Constructor`](braid-groups-element-creation.md#example-ex-ddc2bb)

- [Working with Elements of a Braid Group](braid-groups-elements.md)

  - [Accessing Information](braid-groups-elements.md#accessing-information)

    - [`Parent(u): GrpBrdElt → GrpBrd`](braid-groups-elements.md#function-grpbrd-parent)

    - [`# u: GrpBrdElt → RngIntElt`](braid-groups-elements.md#operation-operation-grpbrdelt-rngintelt)

    - [`CanonicalFactorRepresentation(u: parameters): GrpBrdElt → Tup`](braid-groups-elements.md#function-grpbrd-canonicalfactorrepresentation)

    - [`CFP(u: parameters): GrpBrdElt → Tup`](braid-groups-elements.md#function-cfp-grpbrdelt)

    - [`WordToSequence(u: parameters): GrpBrdElt → SeqEnum`](braid-groups-elements.md#function-wordtosequence-grpbrdelt)

    - [`ElementToSequence(u: parameters): GrpBrdElt → SeqEnum`](braid-groups-elements.md#function-elementtosequence-grpbrdelt)

    - [`Eltseq(u: parameters): GrpBrdElt → SeqEnum`](braid-groups-elements.md#function-eltseq-grpbrdelt)

    - [`InducedPermutation(u): GrpBrdElt → GrpPermElt`](braid-groups-elements.md#function-inducedpermutation-grpbrdelt)

    - [`CanonicalLength(u: parameters): GrpBrdElt → RngIntElt`](braid-groups-elements.md#function-grpbrd-canonicallength)

    - [`Infimum(u: parameters): GrpBrdElt → RngIntElt`](braid-groups-elements.md#function-grpbrd-infimum)

    - [`Supremum(u: parameters): GrpBrdElt → RngIntElt`](braid-groups-elements.md#function-grpbrd-supremum)

    - [`SuperSummitCanonicalLength(u: parameters): GrpBrdElt → RngIntElt`](braid-groups-elements.md#function-grpbrd-supersummitcanonicallength)

    - [`SuperSummitInfimum(u: parameters): GrpBrdElt → RngIntElt`](braid-groups-elements.md#function-grpbrd-supersummitinfimum)

    - [`SuperSummitSupremum(u: parameters): GrpBrdElt → RngIntElt`](braid-groups-elements.md#function-grpbrd-supersummitsupremum)

    - [`Example: Access`](braid-groups-elements.md#example-ex-0cc874)

  - [Computing Normal Forms of Elements](braid-groups-elements.md#computing-normal-forms-of-elements)

    - [`LeftNormalForm(u: parameters): GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-leftnormalform-grpbrdelt)

    - [`NormalForm(u: parameters): GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-grpbrd-normalform)

    - [`LeftNormalForm(~u: parameters): GrpBrdElt`](braid-groups-elements.md#function-leftnormalform-grpbrdelt-ref)

    - [`NormalForm(~u: parameters): GrpBrdElt`](braid-groups-elements.md#function-normalform-grpbrdelt-ref)

    - [`RightNormalForm(u: parameters): GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-rightnormalform-grpbrdelt)

    - [`RightNormalForm(~u: parameters): GrpBrdElt`](braid-groups-elements.md#function-rightnormalform-grpbrdelt-ref)

    - [`LeftMixedCanonicalForm(u: parameters): GrpBrdElt → Tup, Tup`](braid-groups-elements.md#function-leftmixedcanonicalform-grpbrdelt)

    - [`MixedCanonicalForm(u: parameters): GrpBrdElt → Tup, Tup`](braid-groups-elements.md#function-mixedcanonicalform-grpbrdelt)

    - [`RightMixedCanonicalForm(u: parameters): GrpBrdElt → Tup, Tup`](braid-groups-elements.md#function-rightmixedcanonicalform-grpbrdelt)

    - [`Example: Normal Form`](braid-groups-elements.md#example-ex-9c32e6)

  - [Arithmetic Operators and Functions for Elements](braid-groups-elements.md#arithmetic-operators-and-functions-for-elements)

    - [`u * v: GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#operation-op-times-grpbrdelt-grpbrdelt)

    - [`u *:= v: GrpBrdElt, GrpBrdElt`](braid-groups-elements.md#operation-operation-grpbrdelt-grpbrdelt)

    - [`u / v: GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#operation-op-div-grpbrdelt-grpbrdelt)

    - [`u /:= v: GrpBrdElt, GrpBrdElt`](braid-groups-elements.md#operation-operation-grpbrdelt-grpbrdelt-2)

    - [`u ^ n: GrpBrdElt, RngIntElt → GrpBrdElt`](braid-groups-elements.md#operation-op-pow-grpbrdelt-rngintelt)

    - [`u ^:= n: GrpBrdElt, RngIntElt`](braid-groups-elements.md#operation-operation-grpbrdelt-rngintelt-2)

    - [`u ^ v: GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#operation-op-pow-grpbrdelt-grpbrdelt)

    - [`u ^:= v: GrpBrdElt, GrpBrdElt`](braid-groups-elements.md#operation-operation-grpbrdelt-grpbrdelt-3)

    - [`Inverse(u): GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-inverse-grpbrdelt)

    - [`Inverse(~u): GrpBrdElt`](braid-groups-elements.md#function-inverse-grpbrdelt-ref)

    - [`LeftConjugate(u, v): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-grpbrd-leftconjugate)

    - [`LeftConjugate(~u, v): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-leftconjugate-grpbrdelt-grpbrdelt-ref)

    - [`LeftDiv(u, v): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-leftdiv-grpbrdelt-grpbrdelt)

    - [`LeftDiv(u, ~v): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-leftdiv-grpbrdelt-grpbrdelt-2)

    - [`Cycle(u: parameters): GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-grpbrd-cycle)

    - [`Cycle(~u: parameters): GrpBrdElt`](braid-groups-elements.md#function-cycle-grpbrdelt-ref)

    - [`Decycle(u: parameters): GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-grpbrd-decycle)

    - [`Decycle(~u: parameters): GrpBrdElt`](braid-groups-elements.md#function-decycle-grpbrdelt-ref)

    - [`Example: Arithmetic`](braid-groups-elements.md#example-ex-be378e)

  - [Boolean Predicates for Elements](braid-groups-elements.md#boolean-predicates-for-elements)

    - [`u in B: GrpBrdElt, GrpBrd → BoolElt`](braid-groups-elements.md#operation-op-in-grpbrdelt-grpbrd)

    - [`u notin B: GrpBrdElt, GrpBrd → BoolElt`](braid-groups-elements.md#operation-op-notin-grpbrdelt-grpbrd)

    - [`IsEmptyWord(u: parameters): GrpBrdElt → BoolElt`](braid-groups-elements.md#function-isemptyword-grpbrdelt)

    - [`AreIdentical(u, v: parameters): GrpBrdElt, GrpBrdElt → BoolElt`](braid-groups-elements.md#function-areidentical-grpbrdelt-grpbrdelt)

    - [`IsSimple(u: parameters): GrpBrdElt → BoolElt`](braid-groups-elements.md#function-issimple-grpbrdelt)

    - [`IsSuperSummitRepresentative(u: parameters): GrpBrdElt → BoolElt`](braid-groups-elements.md#function-issupersummitrepresentative-grpbrdelt)

    - [`IsUltraSummitRepresentative(u: parameters): GrpBrdElt → BoolElt`](braid-groups-elements.md#function-isultrasummitrepresentative-grpbrdelt)

    - [`IsIdentity(u: parameters): GrpBrdElt → BoolElt`](braid-groups-elements.md#function-isidentity-grpbrdelt)

    - [`IsId(u: parameters): GrpBrdElt → BoolElt`](braid-groups-elements.md#function-isid-grpbrdelt)

    - [`u eq v: GrpBrdElt, GrpBrdElt → BoolElt`](braid-groups-elements.md#operation-grpbrd-eq)

    - [`u ne v: GrpBrdElt, GrpBrdElt → BoolElt`](braid-groups-elements.md#operation-op-ne-grpbrdelt-grpbrdelt)

    - [`u le v: GrpBrdElt, GrpBrdElt → BoolElt`](braid-groups-elements.md#operation-grpbrd-le)

    - [`IsLE(u, v: parameters): GrpBrdElt, GrpBrdElt → BoolElt`](braid-groups-elements.md#function-isle-grpbrdelt-grpbrdelt)

    - [`IsLe(u, v: parameters): GrpBrdElt, GrpBrdElt → BoolElt`](braid-groups-elements.md#function-isle-grpbrdelt-grpbrdelt-2)

    - [`u ge v: GrpBrdElt, GrpBrdElt → BoolElt`](braid-groups-elements.md#operation-grpbrd-ge)

    - [`IsGE(u, v: parameters): GrpBrdElt, GrpBrdElt → BoolElt`](braid-groups-elements.md#function-isge-grpbrdelt-grpbrdelt)

    - [`IsGe(u, v: parameters): GrpBrdElt, GrpBrdElt → BoolElt`](braid-groups-elements.md#function-isge-grpbrdelt-grpbrdelt-2)

    - [`IsConjugate(u, v: parameters): GrpBrdElt, GrpBrdElt → BoolElt, GrpBrdElt`](braid-groups-elements.md#function-grpbrd-isconjugate)

    - [`Example: Boolean`](braid-groups-elements.md#example-ex-58fe4b)

  - [Lattice Operations](braid-groups-elements.md#lattice-operations)

    - [`LeftGCD(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-leftgcd-grpbrdelt-grpbrdelt)

    - [`LeftGcd(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-leftgcd-grpbrdelt-grpbrdelt-2)

    - [`LeftGreatestCommonDivisor(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-leftgreatestcommondivisor-grpbrdelt-grpbrdelt)

    - [`GCD(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-gcd-grpbrdelt-grpbrdelt)

    - [`Gcd(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-gcd-grpbrdelt-grpbrdelt-2)

    - [`GreatestCommonDivisor(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-greatestcommondivisor-grpbrdelt-grpbrdelt)

    - [`RightGCD(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-rightgcd-grpbrdelt-grpbrdelt)

    - [`RightGcd(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-rightgcd-grpbrdelt-grpbrdelt-2)

    - [`RightGreatestCommonDivisor(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-rightgreatestcommondivisor-grpbrdelt-grpbrdelt)

    - [`LeftGCD(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-leftgcd-setq)

    - [`LeftGcd(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-leftgcd-setq-2)

    - [`LeftGreatestCommonDivisor(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-leftgreatestcommondivisor-setq)

    - [`GCD(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-gcd-setq)

    - [`Gcd(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-gcd-setq-2)

    - [`GreatestCommonDivisor(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-greatestcommondivisor-setq)

    - [`RightGCD(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-rightgcd-setq)

    - [`RightGcd(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-rightgcd-setq-2)

    - [`RightGreatestCommonDivisor(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-rightgreatestcommondivisor-setq)

    - [`LeftLCM(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-leftlcm-grpbrdelt-grpbrdelt)

    - [`LeftLcm(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-leftlcm-grpbrdelt-grpbrdelt-2)

    - [`LeftLeastCommonMultiple(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-leftleastcommonmultiple-grpbrdelt-grpbrdelt)

    - [`LCM(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-lcm-grpbrdelt-grpbrdelt)

    - [`Lcm(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-lcm-grpbrdelt-grpbrdelt-2)

    - [`LeastCommonMultiple(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-leastcommonmultiple-grpbrdelt-grpbrdelt)

    - [`RightLCM(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-rightlcm-grpbrdelt-grpbrdelt)

    - [`RightLcm(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-rightlcm-grpbrdelt-grpbrdelt-2)

    - [`RightLeastCommonMultiple(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-rightleastcommonmultiple-grpbrdelt-grpbrdelt)

    - [`LeftLCM(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-leftlcm-setq)

    - [`LeftLcm(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-leftlcm-setq-2)

    - [`LeftLeastCommonMultiple(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-leftleastcommonmultiple-setq)

    - [`LCM(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-lcm-setq)

    - [`Lcm(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-lcm-setq-2)

    - [`LeastCommonMultiple(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-leastcommonmultiple-setq)

    - [`RightLCM(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-rightlcm-setq)

    - [`RightLcm(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-rightlcm-setq-2)

    - [`RightLeastCommonMultiple(S: parameters): Setq → GrpBrdElt`](braid-groups-elements.md#function-rightleastcommonmultiple-setq)

    - [`Example: Boolean`](braid-groups-elements.md#example-ex-3e6267)

  - [Invariants of Conjugacy Classes](braid-groups-elements.md#invariants-of-conjugacy-classes)

    - [`PositiveConjugates(u: parameters): GrpBrdElt → SetIndx`](braid-groups-elements.md#function-grpbrd-positiveconjugates)

    - [`SuperSummitRepresentative(u: parameters): GrpBrdElt → GrpBrdElt, GrpBrdElt`](braid-groups-elements.md#function-supersummitrepresentative-grpbrdelt)

    - [`SuperSummitSet(u: parameters): GrpBrdElt → SetIndx`](braid-groups-elements.md#function-grpbrd-supersummitset)

    - [`UltraSummitRepresentative(u: parameters): GrpBrdElt → GrpBrdElt, GrpBrdElt`](braid-groups-elements.md#function-ultrasummitrepresentative-grpbrdelt)

    - [`UltraSummitSet(u: parameters): GrpBrdElt → SetIndx`](braid-groups-elements.md#function-grpbrd-ultrasummitset)

    - [`Example: Conjugates`](braid-groups-elements.md#example-ex-ea29be)

    - [Computing Class Invariants Interactively](braid-groups-elements.md#computing-class-invariants-interactively)

      - [`PositiveConjugatesProcess(u: parameters): GrpBrdElt → GrpBrdClassProc`](braid-groups-elements.md#function-grpbrd-positiveconjugatesprocess)

      - [`SuperSummitProcess(u: parameters): GrpBrdElt → GrpBrdClassProc`](braid-groups-elements.md#function-grpbrd-supersummitprocess)

      - [`UltraSummitProcess(u: parameters): GrpBrdElt → GrpBrdClassProc`](braid-groups-elements.md#function-grpbrd-ultrasummitprocess)

      - [`BaseElement(P): GrpBrdClassProc → GrpBrdElt`](braid-groups-elements.md#function-baseelement-grpbrdclassproc)

      - [`# P: GrpBrdClassProc → RngIntElt`](braid-groups-elements.md#operation-operation-grpbrdclassproc-rngintelt)

      - [`Representative(P): GrpBrdClassProc → GrpBrdElt`](braid-groups-elements.md#function-grpbrd-representative)

      - [`Rep(P): GrpBrdClassProc → GrpBrdElt`](braid-groups-elements.md#function-rep-grpbrdclassproc)

      - [`IsEmpty(P): GrpBrdClassProc → BoolElt`](braid-groups-elements.md#function-grpbrd-isempty)

      - [`Elements(P): GrpBrdClassProc → SetIndx`](braid-groups-elements.md#function-grpbrd-elements)

      - [`u in P: GrpBrdElt, GrpBrdClassProc → BoolElt, GrpBrdElt`](braid-groups-elements.md#operation-op-in-grpbrdelt-grpbrdclassproc)

      - [`u notin P: GrpBrdElt, GrpBrdClassProc → BoolElt`](braid-groups-elements.md#operation-op-notin-grpbrdelt-grpbrdclassproc)

      - [`NextElement(~P): GrpBrdClassProc`](braid-groups-elements.md#function-nextelement-grpbrdclassproc-ref)

      - [`Complete(~P): GrpBrdClassProc`](braid-groups-elements.md#function-complete-grpbrdclassproc-ref)

      - [`Example: Conjugates Process`](braid-groups-elements.md#example-ex-a3dd2a)

    - [Computing Minimal Simple Elements](braid-groups-elements.md#computing-minimal-simple-elements)

      - [`MinimalElementConjugatingToPositive(x, s: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-minimalelementconjugatingtopositive-grpbrdelt-grpbrdelt)

      - [`MinimalElementConjugatingToSuperSummit(x, s: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-minimalelementconjugatingtosupersummit-grpbrdelt-grpbrdelt)

      - [`MinimalElementConjugatingToUltraSummit(x, s: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-minimalelementconjugatingtoultrasummit-grpbrdelt-grpbrdelt)

      - [`Transport(x, s: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-grpbrd-transport)

      - [`Pullback(x, s: parameters): GrpBrdElt, GrpBrdElt → GrpBrdElt`](braid-groups-elements.md#function-grpbrd-pullback)

      - [`Example: Minimal Simple Elements`](braid-groups-elements.md#example-ex-3ccfd3)

- [Homomorphisms](homomorphism.md)

  - [General Remarks](homomorphism.md#general-remarks)

  - [Constructing Homomorphisms](homomorphism.md#constructing-homomorphisms)

    - [`hom< B -> G | S : parameters >: Struct, Struct → Map`](homomorphism.md#constructor-constructor-hom-struct-struct-map)

  - [Accessing Homomorphisms](homomorphism.md#accessing-homomorphisms)

    - [`e @ f: GrpBrdElt, Map → GrpElt`](homomorphism.md#literal-literal-e-f-grpbrdelt-map-grpelt)

    - [`f(e): Map, GrpBrdElt → GrpElt`](homomorphism.md#literal-f-map-grpbrdelt)

    - [`B @ f: GrpBrd, Map → Grp`](homomorphism.md#literal-literal-b-f-grpbrd-map-grp)

    - [`f(B): Map, GrpBrd → Grp`](homomorphism.md#literal-f-map-grpbrd)

    - [`u @@ f: GrpElt, Map → GrpBrdElt`](homomorphism.md#literal-literal-u-f-grpelt-map-grpbrdelt)

    - [`Domain(f): Map → Grp`](homomorphism.md#function-domain-map)

    - [`Codomain(f): Map → Grp`](homomorphism.md#function-codomain-map)

    - [`Image(f): Map → Grp`](homomorphism.md#function-image-map)

    - [`Example: Homomorphisms`](homomorphism.md#example-ex-5eaf4d)

  - [Representations of Braid Groups](homomorphism.md#representations-of-braid-groups)

    - [`SymmetricRepresentation(B): GrpBrd → Map`](homomorphism.md#function-symmetricrepresentation-grpbrd)

    - [`BurauRepresentation(B): GrpBrd → Map`](homomorphism.md#function-buraurepresentation-grpbrd)

    - [`BurauRepresentation(B, p): GrpBrd, RngIntElt → Map`](homomorphism.md#function-buraurepresentation-grpbrd-rngintelt)

    - [`Example: Representations`](homomorphism.md#example-ex-14a9f2)