Braid Groups
- Introduction
- Constructing and Accessing Braid Groups
- Creating Elements of a Braid Group
Representative(B): GrpBrd → GrpBrdEltRep(B): GrpBrd → GrpBrdEltIdentity(B): GrpBrd → GrpBrdEltId(B): GrpBrd → GrpBrdEltB ! 1: GrpBrd, RngIntElt → GrpBrdEltFundamentalElement(B: parameters): GrpBrd → GrpBrdEltGenerators(B: parameters): GrpBrd → [ GrpBrd ]B . i: GrpBrd, RngIntElt → GrpBrdEltB . T: GrpBrd, Tup → GrpBrdEltB ! [ i₁, ..., iₖ ]: GrpBrd, [ RngIntElt ] → GrpBrdEltB ! [ T₁, ..., Tₖ ]: GrpBrd, [ Tup ] → GrpBrdEltB ! p: GrpBrd, GrpPermElt → GrpBrdEltB ! [ p₁, ...,pₖ ]: GrpBrd, [ GrpPermElt ] → GrpBrdEltB ! T: GrpBrd, Tup → GrpBrdEltIsProductOfParallelDescendingCycles(p): GrpPermElt → BoolEltRandom(B, r, s, m, n: parameters): GrpBrd, RngIntElt, RngIntElt, RngIntElt, RngIntElt → GrpBrdEltRandomCFP(B, r, s, m, n: parameters): GrpBrd, RngIntElt, RngIntElt → GrpBrdEltRandom(B: parameters): GrpBrd → GrpBrdEltRandomCFP(B: parameters): GrpBrd → GrpBrdEltRandom(B, m, n: parameters): GrpBrd, RngIntElt, RngIntElt → GrpBrdEltRandomWord(B, m, n: parameters): GrpBrd, RngIntElt, RngIntElt → GrpBrdEltRandomWord(B: parameters): GrpBrd → GrpBrdEltExample: Constructor
- Working with Elements of a Braid Group
- Accessing Information
- Computing Normal Forms of Elements
- Arithmetic Operators and Functions for Elements
u * v: GrpBrdElt, GrpBrdElt → GrpBrdEltu *:= v: GrpBrdElt, GrpBrdEltu / v: GrpBrdElt, GrpBrdElt → GrpBrdEltu /:= v: GrpBrdElt, GrpBrdEltu ^ n: GrpBrdElt, RngIntElt → GrpBrdEltu ^:= n: GrpBrdElt, RngIntEltu ^ v: GrpBrdElt, GrpBrdElt → GrpBrdEltu ^:= v: GrpBrdElt, GrpBrdEltInverse(u): GrpBrdElt → GrpBrdEltInverse(~u): GrpBrdEltLeftConjugate(u, v): GrpBrdElt, GrpBrdElt → GrpBrdEltLeftConjugate(~u, v): GrpBrdElt, GrpBrdElt → GrpBrdEltLeftDiv(u, v): GrpBrdElt, GrpBrdElt → GrpBrdEltLeftDiv(u, ~v): GrpBrdElt, GrpBrdElt → GrpBrdEltCycle(u: parameters): GrpBrdElt → GrpBrdEltCycle(~u: parameters): GrpBrdEltDecycle(u: parameters): GrpBrdElt → GrpBrdEltDecycle(~u: parameters): GrpBrdEltExample: Arithmetic
- Boolean Predicates for Elements
u in B: GrpBrdElt, GrpBrd → BoolEltu notin B: GrpBrdElt, GrpBrd → BoolEltIsEmptyWord(u: parameters): GrpBrdElt → BoolEltAreIdentical(u, v: parameters): GrpBrdElt, GrpBrdElt → BoolEltIsSimple(u: parameters): GrpBrdElt → BoolEltIsSuperSummitRepresentative(u: parameters): GrpBrdElt → BoolEltIsUltraSummitRepresentative(u: parameters): GrpBrdElt → BoolEltIsIdentity(u: parameters): GrpBrdElt → BoolEltIsId(u: parameters): GrpBrdElt → BoolEltu eq v: GrpBrdElt, GrpBrdElt → BoolEltu ne v: GrpBrdElt, GrpBrdElt → BoolEltu le v: GrpBrdElt, GrpBrdElt → BoolEltIsLE(u, v: parameters): GrpBrdElt, GrpBrdElt → BoolEltIsLe(u, v: parameters): GrpBrdElt, GrpBrdElt → BoolEltu ge v: GrpBrdElt, GrpBrdElt → BoolEltIsGE(u, v: parameters): GrpBrdElt, GrpBrdElt → BoolEltIsGe(u, v: parameters): GrpBrdElt, GrpBrdElt → BoolEltIsConjugate(u, v: parameters): GrpBrdElt, GrpBrdElt → BoolElt, GrpBrdEltExample: Boolean
- Lattice Operations
LeftGCD(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltLeftGcd(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltLeftGreatestCommonDivisor(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltGCD(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltGcd(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltGreatestCommonDivisor(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltRightGCD(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltRightGcd(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltRightGreatestCommonDivisor(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltLeftGCD(S: parameters): Setq → GrpBrdEltLeftGcd(S: parameters): Setq → GrpBrdEltLeftGreatestCommonDivisor(S: parameters): Setq → GrpBrdEltGCD(S: parameters): Setq → GrpBrdEltGcd(S: parameters): Setq → GrpBrdEltGreatestCommonDivisor(S: parameters): Setq → GrpBrdEltRightGCD(S: parameters): Setq → GrpBrdEltRightGcd(S: parameters): Setq → GrpBrdEltRightGreatestCommonDivisor(S: parameters): Setq → GrpBrdEltLeftLCM(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltLeftLcm(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltLeftLeastCommonMultiple(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltLCM(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltLcm(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltLeastCommonMultiple(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltRightLCM(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltRightLcm(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltRightLeastCommonMultiple(u, v: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltLeftLCM(S: parameters): Setq → GrpBrdEltLeftLcm(S: parameters): Setq → GrpBrdEltLeftLeastCommonMultiple(S: parameters): Setq → GrpBrdEltLCM(S: parameters): Setq → GrpBrdEltLcm(S: parameters): Setq → GrpBrdEltLeastCommonMultiple(S: parameters): Setq → GrpBrdEltRightLCM(S: parameters): Setq → GrpBrdEltRightLcm(S: parameters): Setq → GrpBrdEltRightLeastCommonMultiple(S: parameters): Setq → GrpBrdEltExample: Boolean
- Invariants of Conjugacy Classes
PositiveConjugates(u: parameters): GrpBrdElt → SetIndxSuperSummitRepresentative(u: parameters): GrpBrdElt → GrpBrdElt, GrpBrdEltSuperSummitSet(u: parameters): GrpBrdElt → SetIndxUltraSummitRepresentative(u: parameters): GrpBrdElt → GrpBrdElt, GrpBrdEltUltraSummitSet(u: parameters): GrpBrdElt → SetIndxExample: Conjugates- Computing Class Invariants Interactively
- Computing Minimal Simple Elements
MinimalElementConjugatingToPositive(x, s: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltMinimalElementConjugatingToSuperSummit(x, s: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltMinimalElementConjugatingToUltraSummit(x, s: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltTransport(x, s: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltPullback(x, s: parameters): GrpBrdElt, GrpBrdElt → GrpBrdEltExample: Minimal Simple Elements
- Homomorphisms
