Title: The Braid Groups
Version: 1.0.0
Description: Deals with the braid groups. Includes creation of some specific braids, group operations, free reduction, and Bronfman polynomials. Braid theory has applications in fluid mechanics and quantum physics. The code is adapted from the 'Haskell' library 'combinat', and is based on Birman and Brendle (2005) <doi:10.48550/arXiv.math/0409205>.
License: GPL-3
URL: https://github.com/stla/braids
BugReports: https://github.com/stla/braids/issues
Imports: maybe
Encoding: UTF-8
RoxygenNote: 7.3.1
NeedsCompilation: no
Packaged: 2024-04-26 16:57:22 UTC; SDL96354
Author: Stéphane Laurent [aut, cre], Balazs Komuves [cph] (author of the Haskell library 'combinat')
Maintainer: Stéphane Laurent <laurent_step@outlook.fr>
Repository: CRAN
Date/Publication: 2024-04-28 10:00:03 UTC

Braid words of given length

Description

All braid words of the given length.

Usage

allBraidWords(n, l)

Arguments

n

number of strands, integer >=2

l

length of the words

Value

A list of braid objects.

Examples

allBraidWords(3, 2)

Positive braid words of given length

Description

All positive braid words of the given length.

Usage

allPositiveBraidWords(n, l)

Arguments

n

number of strands, integer >=2

l

length of the words

Value

A list of braid objects.

Examples

allPositiveBraidWords(3, 4)

ASCII braid

Description

Prints an ASCII figure of a braid.

Usage

braidASCII(braid)

Arguments

braid

a braid object

Value

No value is returned, just prints the ASCII figure.

Examples

braid <- mkBraid(4, c(1, -2))
braidASCII(braid)

Braid permutation

Description

Returns the left-to-right permutation associated to a braid.

Usage

braidPermutation(braid)

Arguments

braid

a braid object (e.g. created with mkBraid)

Value

A permutation.

Examples

braid <- mkBraid(4, c(2, -3, 3))
braidPermutation(braid)

Bronfman polynomials

Description

The Bronfman polynomial of a braid group is the reciprocal of the growth function of the positive braids. This function computes the Bronfman polynomial of the braid group on n strands for n going to 1 to N.

Usage

bronfmanPolynomials(N)

Arguments

N

maximum number of strands

Value

A list of integer vectors representing the Bronfman polynomials; each vector represents the polynomial coefficients in increasing order.

Examples

bronfmanPolynomials(3) # 1, 1 - X, 1 - 2X + X^3

Composition of many braids.

Description

Composes many braids, doing free reduction on the result.

Usage

composeManyBraids(braids)

Arguments

braids

list of braid objects with the same number of strands

Value

A braid object.

Examples

braid <- mkBraid(4, c(2, -3, 3))
composeManyBraids(list(braid, braid, braid))

Composition of two braids

Description

Composes two braids, doing free reduction on the result.

Usage

composeTwoBraids(braid1, braid2)

Arguments

braid1, braid2

braid objects with the same number of strands

Value

A braid object.

Examples

braid <- mkBraid(4, c(2, -3, 3))
composeTwoBraids(braid, braid)

Double generator

Description

Generator \sigma_{s,t} in the Birman-Ko-Lee new presentation. It twists the strands s and t while going over all other strands (for t=s+1, this is \sigma_s).

Usage

doubleSigma(n, s, t)

Arguments

n

number of strands, integer >=2

s, t

indices of two strands, s < t

Value

A braid object.

Examples

doubleSigma(5, 1, 3)

Free reduction of a braid

Description

Applies free reduction to a braid, i.e. removes pairs of consecutive generators inverse of each other.

Usage

freeReduceBraidWord(braid)

Arguments

braid

a braid object (e.g. created with mkBraid)

Value

A braid object.

Examples

braid <- mkBraid(4, c(2, -3, 3))
freeReduceBraidWord(braid)

Half-twist

Description

The (positive) half-twist of all the braid strands, usually denoted by \Delta.

Usage

halfTwist(n)

Arguments

n

number of strands, integer >=2

Value

A braid object.

Examples

halfTwist(4)

Inverse braid

Description

The inverse of a braid (without performing reduction).

Usage

inverseBraid(braid)

Arguments

braid

a braid object

Value

A braid object.

Examples

braid <- mkBraid(4, c(2, -3, 3))
ibraid <- inverseBraid(braid)
composeTwoBraids(braid, ibraid)

Whether a braid is a permutation braid

Description

Checks whether a braid is a permutation braid, that is, a positive braid where any two strands cross at most one, and positively.

Usage

isPermutationBraid(braid)

Arguments

braid

a braid object

Value

A Boolean value.

Examples

braid <- mkBraid(4, c(2, -3, 3))
isPermutationBraid(braid)

Whether a braid is positive

Description

Checks whether a braid has only positive Artin generators.

Usage

isPositiveBraidWord(braid)

Arguments

braid

a braid object

Value

A Boolean value.

Examples

braid <- mkBraid(4, c(2, -3, 3))
isPositiveBraidWord(braid)

Whether a braid is pure

Description

Checks whether a braid is pure, i.e. its permutation is trivial.

Usage

isPureBraid(braid)

Arguments

braid

a braid object

Value

A Boolean value.

Examples

braid <- mkBraid(4, c(2, -3, 3))
isPureBraid(braid)

Linking matrix

Description

Linking numbers between all pairs of strands of a braid.

Usage

linkingMatrix(braid)

Arguments

braid

a braid object

Value

A matrix.

See Also

strandLinking to get the linking number between two strands of the braid.

Examples

braid <- mkBraid(4, c(2, -3, 3))
linkingMatrix(braid)

Make a braid

Description

Make a braid.

Usage

mkBraid(n, artingens)

Arguments

n

number of strands, an integer, at least 2

artingens

Artin generators given by a vector of non-zero integers; a positive integer i corresponds to the standard positive Artin generator of a braid which represents twisting the neighbour strands i and i+1, such that strand i goes under strand i+1; a negative integer -i corresponds to the inverse.

Value

A braid object.

Examples

mkBraid(n = 4, c(2, -3))

Number of strands

Description

The number of strands of a braid.

Usage

numberOfStrands(braid)

Arguments

braid

a braid object (e.g. created with mkBraid)

Value

An integer.

Permutation braid

Description

Makes a permutation braid from a permutation.

Usage

permutationBraid(perm)

Arguments

perm

a permutation

Value

A braid object.

Examples

perm <- c(3, 1, 4, 2)
braid <- permutationBraid(perm)
isPermutationBraid(braid)
braidPermutation(braid)

Linking number between two strands

Description

The linking number between two strands of a braid.

Usage

strandLinking(braid, i, j)

Arguments

braid

a braid object

i, j

indices of two strands

Value

An integer.

See Also

linkingMatrix to get the linking numbers between all pairs of strands of the braid.

Examples

braid <- mkBraid(4, c(2, -3, 3))
strandLinking(braid, 1, 3)

Inner automorphism

Description

The inner automorphism defined by \tau X = \Delta^{-1} X \Delta, where \Delta is the positive half-twist; it sends each generator \sigma_j to \sigma_{n-j}.

Usage

tau(braid)

Arguments

braid

a braid object

Value

A braid object.

Examples

braid <- mkBraid(4, c(2, -3, 3))
tau(braid)