Package 'braids'

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)

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