BRAID GROUPS
Acknowledgements
Introduction
Lattice Structure and Simple Elements
Representing Elements of a Braid Group
Automatic Conversions
Default Presentations
Representation Used for Group Operations
Printing of Elements
Normal Form for Elements of a Braid Group
Mixed Canonical Form and Lattice Operations
Conjugacy Testing and Conjugacy Search
Definition of the Class Invariants
Computing the Class Invariants
Conjugacy Testing and Conjugacy Search
Constructing and Accessing Braid Groups
Creating Elements of a Braid Group
Working with Elements of a Braid Group
Accessing Information
Computing Normal Forms of Elements
Arithmetic Operators and Functions for Elements
Boolean Predicates for Elements
Lattice Operations
Invariants of Conjugacy Classes
Computing Class Invariants Interactively
Computing Minimal Simple Elements
Homomorphisms
General Remarks
Constructing Homomorphisms
Accessing Homomorphisms
Representations of Braid Groups
Bibliography
Introduction
Lattice Structure and Simple Elements
Representing Elements of a Braid Group
Automatic Conversions
Default Presentations
Representation Used for Group Operations
Printing of Elements
Normal Form for Elements of a Braid Group
Mixed Canonical Form and Lattice Operations
Conjugacy Testing and Conjugacy Search
Definition of the Class Invariants
Computing the Class Invariants
Conjugacy Testing and Conjugacy Search
Constructing and Accessing Braid Groups
BraidGroup(n: parameters) : RngIntElt -> GrpBrd
GetPresentation(B) : GrpBrd -> MonStgElt
SetPresentation(~B, s) : GrpBrd, MonStgElt ->
GetForceCFP(B) : GrpBrd -> BoolElt
SetForceCFP(~B, b) : GrpBrd, BoolElt ->
GetElementPrintFormat(B) : GrpBrd -> MonStgElt
SetElementPrintFormat(~B, s) : GrpBrd, MonStgElt ->
NumberOfStrings(B) : GrpBrd -> RngIntElt
NumberOfGenerators(B) : GrpBrd -> RngIntElt
Creating Elements of a Braid Group
Representative(B) : GrpBrd -> GrpBrdElt
Identity(B) : GrpBrd -> GrpBrdElt
FundamentalElement(B: parameters) : GrpBrd -> GrpBrdElt
Generators(B: parameters) : GrpBrd -> [ GrpBrd ]
B . i : GrpBrd, RngIntElt -> GrpBrdElt
B . T : GrpBrd, Tup -> GrpBrdElt
B ! [ i1, ..., ik ] : GrpBrd, [ RngIntElt ] -> GrpBrdElt
B ! [ T1, ..., Tk ] : GrpBrd, [ Tup ] -> GrpBrdElt
B p : GrpBrd, GrpPermElt -> GrpBrdElt
B ! [ p1, ...,pk ]: GrpBrd, [ GrpPermElt ] -> GrpBrdElt
B T : GrpBrd, Tup -> GrpBrdElt
IsProductOfParallelDescendingCycles(p) : GrpPermElt -> BoolElt
Random(B, r, s, m, n: parameters) : GrpBrd, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> GrpBrdElt
Random(B, m, n: parameters) : GrpBrd, RngIntElt, RngIntElt -> GrpBrdElt
Example GrpBrd_Constructor (H82E1)
Working with Elements of a Braid Group
Accessing Information
Parent(u) : GrpBrdElt -> GrpBrd
# u : GrpBrdElt -> RngIntElt
CanonicalFactorRepresentation(u: parameters) : GrpBrdElt -> Tup
WordToSequence(u: parameters) : GrpBrdElt -> SeqEnum
InducedPermutation(u) : GrpBrdElt -> GrpPermElt
CanonicalLength(u: parameters) : GrpBrdElt -> RngIntElt
Infimum(u: parameters) : GrpBrdElt -> RngIntElt
Supremum(u: parameters) : GrpBrdElt -> RngIntElt
SuperSummitCanonicalLength(u: parameters) : GrpBrdElt -> RngIntElt
SuperSummitInfimum(u: parameters) : GrpBrdElt -> RngIntElt
SuperSummitSupremum(u: parameters) : GrpBrdElt -> RngIntElt
Example GrpBrd_Access (H82E2)
Computing Normal Forms of Elements
LeftNormalForm(u: parameters) : GrpBrdElt -> GrpBrdElt
LeftNormalForm(~u: parameters) : GrpBrdElt ->
RightNormalForm(u: parameters) : GrpBrdElt -> GrpBrdElt
RightNormalForm(~u: parameters) : GrpBrdElt ->
LeftMixedCanonicalForm(u: parameters) : GrpBrdElt -> Tup, Tup
RightMixedCanonicalForm(u: parameters) : GrpBrdElt -> Tup, Tup
Example GrpBrd_NormalForm (H82E3)
Arithmetic Operators and Functions for Elements
u * v : GrpBrdElt, GrpBrdElt -> GrpBrdElt
u *:= v : GrpBrdElt, GrpBrdElt ->
u / v : GrpBrdElt, GrpBrdElt -> GrpBrdElt
u /:= v : GrpBrdElt, GrpBrdElt ->
u ^ n : GrpBrdElt, RngIntElt -> GrpBrdElt
u ^:= n : GrpBrdElt, RngIntElt ->
u ^ v : GrpBrdElt, GrpBrdElt -> GrpBrdElt
u ^:= v : GrpBrdElt, GrpBrdElt ->
Inverse(u) : GrpBrdElt -> GrpBrdElt
Inverse(~u) : GrpBrdElt ->
LeftConjugate(u, v) : GrpBrdElt, GrpBrdElt -> GrpBrdElt
LeftConjugate(~u, v) : GrpBrdElt, GrpBrdElt -> GrpBrdElt
LeftDiv(u, v) : GrpBrdElt, GrpBrdElt -> GrpBrdElt
LeftDiv(u, ~v) : GrpBrdElt, GrpBrdElt -> GrpBrdElt
Cycle(u: parameters) : GrpBrdElt -> GrpBrdElt
Cycle(~u: parameters) : GrpBrdElt ->
Decycle(u: parameters) : GrpBrdElt -> GrpBrdElt
Decycle(~u: parameters) : GrpBrdElt ->
Example GrpBrd_Arithmetic (H82E4)
Boolean Predicates for Elements
u in B : GrpBrdElt, GrpBrd -> BoolElt
u notin B : GrpBrdElt, GrpBrd -> BoolElt
IsEmptyWord(u: parameters) : GrpBrdElt -> BoolElt
AreIdentical(u, v: parameters) : GrpBrdElt, GrpBrdElt -> BoolElt
IsSimple(u: parameters) : GrpBrdElt -> BoolElt
IsSuperSummitRepresentative(u: parameters) : GrpBrdElt -> BoolElt
IsUltraSummitRepresentative(u: parameters) : GrpBrdElt -> BoolElt
IsIdentity(u: parameters) : GrpBrdElt -> BoolElt
u eq v : GrpBrdElt, GrpBrdElt -> BoolElt
u ne v : GrpBrdElt, GrpBrdElt -> BoolElt
u ≤v : GrpBrdElt, GrpBrdElt -> BoolElt
u ≥v : GrpBrdElt, GrpBrdElt -> BoolElt
IsConjugate(u, v: parameters) : GrpBrdElt, GrpBrdElt -> BoolElt, GrpBrdElt
Example GrpBrd_Boolean (H82E5)
Lattice Operations
LeftGCD(u, v: parameters) : GrpBrdElt, GrpBrdElt -> GrpBrdElt
RightGCD(u, v: parameters) : GrpBrdElt, GrpBrdElt -> GrpBrdElt
LeftGCD(S: parameters) : Setq -> GrpBrdElt
RightGCD(S: parameters) : Setq -> GrpBrdElt
LeftLCM(u, v: parameters) : GrpBrdElt, GrpBrdElt -> GrpBrdElt
RightLCM(u, v: parameters) : GrpBrdElt, GrpBrdElt -> GrpBrdElt
LeftLCM(S: parameters) : Setq -> GrpBrdElt
RightLCM(S: parameters) : Setq -> GrpBrdElt
Example GrpBrd_Boolean (H82E6)
Invariants of Conjugacy Classes
PositiveConjugates(u: parameters) : GrpBrdElt -> SetIndx
SuperSummitRepresentative(u: parameters) : GrpBrdElt -> GrpBrdElt, GrpBrdElt
SuperSummitSet(u: parameters) : GrpBrdElt -> SetIndx
UltraSummitRepresentative(u: parameters) : GrpBrdElt -> GrpBrdElt, GrpBrdElt
UltraSummitSet(u: parameters) : GrpBrdElt -> SetIndx
Example GrpBrd_Conjugates (H82E7)
Computing Class Invariants Interactively
PositiveConjugatesProcess(u: parameters) : GrpBrdElt -> GrpBrdClassProc
SuperSummitProcess(u: parameters) : GrpBrdElt -> GrpBrdClassProc
UltraSummitProcess(u: parameters) : GrpBrdElt -> GrpBrdClassProc
BaseElement(P) : GrpBrdClassProc -> GrpBrdElt
# P : GrpBrdClassProc -> RngIntElt
Representative(P) : GrpBrdClassProc -> GrpBrdElt
IsEmpty(P) : GrpBrdClassProc -> BoolElt
Elements(P) : GrpBrdClassProc -> SetIndx
u in P : GrpBrdElt, GrpBrdClassProc -> BoolElt, GrpBrdElt
u notin P : GrpBrdElt, GrpBrdClassProc -> BoolElt
NextElement(~P) : GrpBrdClassProc ->
Complete(~P) : GrpBrdClassProc ->
Example GrpBrd_ConjugatesProcess (H82E8)
Computing Minimal Simple Elements
MinimalElementConjugatingToPositive(x, s: parameters) : GrpBrdElt, GrpBrdElt -> GrpBrdElt
MinimalElementConjugatingToSuperSummit(x, s: parameters) : GrpBrdElt, GrpBrdElt -> GrpBrdElt
MinimalElementConjugatingToUltraSummit(x, s: parameters) : GrpBrdElt, GrpBrdElt -> GrpBrdElt
Transport(x, s: parameters) : GrpBrdElt, GrpBrdElt -> GrpBrdElt
Pullback(x, s: parameters) : GrpBrdElt, GrpBrdElt -> GrpBrdElt
Example GrpBrd_MinimalSimpleElements (H82E9)
Homomorphisms
General Remarks
Constructing Homomorphisms
hom< B -> G | S : parameters > : Struct , Struct -> Map
Accessing Homomorphisms
e @ f : GrpBrdElt, Map -> GrpElt
B @ f : GrpBrd, Map -> Grp
u @@ f : GrpElt, Map -> GrpBrdElt
Domain(f) : Map -> Grp
Codomain(f) : Map -> Grp
Image(f) : Map -> Grp
Example GrpBrd_Homomorphisms (H82E10)
Representations of Braid Groups
SymmetricRepresentation(B) : GrpBrd -> Map
BurauRepresentation(B) : GrpBrd -> Map
BurauRepresentation(B, p) : GrpBrd, RngIntElt -> Map
Example GrpBrd_Representations (H82E11)
Bibliography
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