BKLRightNormalForm Class Reference

(Contains errors!!! Not to be used yet) Birman-Ko-Lee right normal form. More...

#include <BKLRightNormalForm.h>

Public Types
typedef triple< int, int, list < Permutation > >	NF
	Presentation of a normal form.
Public Member Functions
	BKLRightNormalForm ()
	Create trivial normal form.
	BKLRightNormalForm (const BKLRightNormalForm &bkl)
	Copy constructor.
	BKLRightNormalForm (int rank, int power, const list< Permutation > &decomp)
	Create normal form by its presentation (no check that given presentation is correct, use adjustDecomposition if the presentation is incorrect).
	BKLRightNormalForm (const BraidGroup &G, const Word &w)
	Create a normal form of a braid word.
BKLRightNormalForm &	operator= (const BKLRightNormalForm &bkl)
	Assignment operator.
BKLRightNormalForm &	operator= (const NF &bkl)
	Assignment operator.
BKLRightNormalForm	operator- () const
	Invert a normal form.
BKLRightNormalForm	operator* (const BKLRightNormalForm &bkl) const
	Multiply two normal forms.
bool	operator== (const BKLRightNormalForm &bkl) const
bool	operator!= (const BKLRightNormalForm &bkl) const
bool	operator< (const BKLRightNormalForm &bkl) const
int	getPower () const
	Get the power of omega.
const list< Permutation > &	getDecomposition () const
	Get the decomposition.
	operator NF () const
	Get the presentation of the normal form.
bool	isTrivial () const
	Check if the normal form os trivial.
Word	getWord () const
	Return a word represented by the normal form.
void	setPower (int p)
	Set the power of $\Delta$ . The result is always a correct normal form.
void	setDecomposition (const list< Permutation > &d)
	Set the decomposition $(\xi_1,\ldots,\xi_m)$ . The result might be an incorrect form (not satisfying some gready conditions of this BKL form).
Static Public Member Functions
static Permutation	getTinyTwistPermutation (int theIndex)
	Returns a cyclic permutation $\delta = (1,2,3,\ldots,n-1,0)$ .
static void	adjustDecomposition (int rank, int &power, list< Permutation > &decomp)
	Function adjusting the decomposition in a normal form.
Private Types
enum	transformationResult { TWO_MULTIPLIERS, ONE_MULTIPLIER, NO_CHANGE }
	The result of the transformation. More...
Private Member Functions
BKLRightNormalForm	inverse () const
	Invert the normal form.
BKLRightNormalForm	multiply (const BKLRightNormalForm &bkl) const
	Multiply the normal form by another normal form on the right.
Static Private Member Functions
static transformationResult	transform (int theIndex, Permutation &p1, Permutation &p2)
	Main function for computing the normal form of the word.
Private Attributes
int	theRank
	The rank of the braid group (number of strands).
int	theOmegaPower
	Power of omega.
list< Permutation >	theDecomposition
	Sequence of permutations.

Detailed Description

(Contains errors!!! Not to be used yet) Birman-Ko-Lee right normal form.

The standard presentation for BKL is a pair $(p,(\xi_1,\ldots,\xi_m))$ , where $p\in \mathbb{Z}$ and each $\xi_i \in S_n$ is a permutation.

Definition at line 37 of file BKLRightNormalForm.h.

Member Typedef Documentation

typedef triple< int , int , list< Permutation > > BKLRightNormalForm::NF

Presentation of a normal form.

The first component specifies the rank of a braid group. The second component specifies the power of the twist. The third component specifies the list of braid permutations.

Definition at line 47 of file BKLRightNormalForm.h.

Member Enumeration Documentation

enum BKLRightNormalForm::transformationResult [private]

The result of the transformation.

Enumerator:

TWO_MULTIPLIERS
ONE_MULTIPLIER
NO_CHANGE

Definition at line 193 of file BKLRightNormalForm.h.

Constructor & Destructor Documentation

BKLRightNormalForm::BKLRightNormalForm ( ) [inline]

Create trivial normal form.

Definition at line 58 of file BKLRightNormalForm.h.

BKLRightNormalForm::BKLRightNormalForm ( const BKLRightNormalForm & bkl ) [inline]

Copy constructor.

Definition at line 66 of file BKLRightNormalForm.h.

BKLRightNormalForm::BKLRightNormalForm	(	int	rank,
		int	power,
		const list< Permutation > &	decomp
	)			`[inline]`

Create normal form by its presentation (no check that given presentation is correct, use adjustDecomposition if the presentation is incorrect).

Definition at line 76 of file BKLRightNormalForm.h.

BKLRightNormalForm::BKLRightNormalForm	(	const BraidGroup &	G,
		const Word &	w
	)

Create a normal form of a braid word.

Member Function Documentation

static void BKLRightNormalForm::adjustDecomposition	(	int	rank,
		int &	power,
		list< Permutation > &	decomp
	)			`[static]`

Function adjusting the decomposition in a normal form.

const list< Permutation >& BKLRightNormalForm::getDecomposition ( ) const [inline]

Get the decomposition.

Definition at line 138 of file BKLRightNormalForm.h.

References theDecomposition.

int BKLRightNormalForm::getPower ( ) const [inline]

Get the power of omega.

Definition at line 134 of file BKLRightNormalForm.h.

References theOmegaPower.

static Permutation BKLRightNormalForm::getTinyTwistPermutation ( int theIndex ) [inline, static]

Returns a cyclic permutation $\delta = (1,2,3,\ldots,n-1,0)$ .

Definition at line 150 of file BKLRightNormalForm.h.

Word BKLRightNormalForm::getWord ( ) const

Return a word represented by the normal form.

BKLRightNormalForm BKLRightNormalForm::inverse ( ) const [private]

Invert the normal form.

Referenced by operator-().

bool BKLRightNormalForm::isTrivial ( ) const [inline]

Check if the normal form os trivial.

Definition at line 146 of file BKLRightNormalForm.h.

References theDecomposition, and theOmegaPower.

BKLRightNormalForm BKLRightNormalForm::multiply ( const BKLRightNormalForm & bkl ) const [private]

Multiply the normal form by another normal form on the right.

Referenced by operator*().

BKLRightNormalForm::operator NF ( ) const [inline]

Get the presentation of the normal form.

Definition at line 142 of file BKLRightNormalForm.h.

References theDecomposition, theOmegaPower, and theRank.

bool BKLRightNormalForm::operator!= ( const BKLRightNormalForm & bkl ) const

BKLRightNormalForm BKLRightNormalForm::operator* ( const BKLRightNormalForm & bkl ) const [inline]

Multiply two normal forms.

Definition at line 117 of file BKLRightNormalForm.h.

References multiply().

BKLRightNormalForm BKLRightNormalForm::operator- ( ) const [inline]

Invert a normal form.

Definition at line 113 of file BKLRightNormalForm.h.

References inverse().

bool BKLRightNormalForm::operator< ( const BKLRightNormalForm & bkl ) const

BKLRightNormalForm& BKLRightNormalForm::operator= ( const NF & bkl ) [inline]

Assignment operator.

Definition at line 106 of file BKLRightNormalForm.h.

References triple< T1, T2, T3 >::first, triple< T1, T2, T3 >::second, theDecomposition, theOmegaPower, theRank, and triple< T1, T2, T3 >::third.

BKLRightNormalForm& BKLRightNormalForm::operator= ( const BKLRightNormalForm & bkl ) [inline]

Assignment operator.

Definition at line 99 of file BKLRightNormalForm.h.

References theDecomposition, theOmegaPower, and theRank.

bool BKLRightNormalForm::operator== ( const BKLRightNormalForm & bkl ) const

void BKLRightNormalForm::setDecomposition ( const list< Permutation > & d ) [inline]

Set the decomposition $(\xi_1,\ldots,\xi_m)$ . The result might be an incorrect form (not satisfying some gready conditions of this BKL form).

Definition at line 182 of file BKLRightNormalForm.h.

References theDecomposition.

void BKLRightNormalForm::setPower ( int p ) [inline]

Set the power of $\Delta$ . The result is always a correct normal form.

Definition at line 179 of file BKLRightNormalForm.h.

References theOmegaPower.

static transformationResult BKLRightNormalForm::transform	(	int	theIndex,
		Permutation &	p1,
		Permutation &	p2
	)			`[static, private]`

Main function for computing the normal form of the word.

Member Data Documentation

list< Permutation > BKLRightNormalForm::theDecomposition [private]

Sequence of permutations.

Definition at line 223 of file BKLRightNormalForm.h.

Referenced by getDecomposition(), isTrivial(), operator NF(), operator=(), and setDecomposition().

int BKLRightNormalForm::theOmegaPower [private]

Power of omega.

Definition at line 219 of file BKLRightNormalForm.h.

Referenced by getPower(), isTrivial(), operator NF(), operator=(), and setPower().

int BKLRightNormalForm::theRank [private]

The rank of the braid group (number of strands).

Definition at line 215 of file BKLRightNormalForm.h.

Referenced by operator NF(), and operator=().

The documentation for this class was generated from the following file:

BraidGroup/include/BKLRightNormalForm.h

BKLRightNormalForm Class Reference

Public Types

Public Member Functions

Static Public Member Functions

Private Types

Private Member Functions