BKLLeftNormalForm Class Reference

Birman-Ko-Lee left normal form. More...

#include <BKLLeftNormalForm.h>

Public Types
typedef triple< int, int, list < Permutation > >	NF
	Presentation of a normal form.
Public Member Functions
	BKLLeftNormalForm ()
	Create trivial normal form.
	BKLLeftNormalForm (const BKLLeftNormalForm &bkl)
	Copy constructor.
	BKLLeftNormalForm (int rank, int p, const list< Permutation > &d)
	Create normal form by its presentation (no check that given presentation is correct, use adjustDecomposition if the presentation is incorrect).
	BKLLeftNormalForm (const NF &bkl)
	Create normal form by its presentation (no check that given presentation is correct, use adjustDecomposition if the presentation is incorrect).
	BKLLeftNormalForm (const BraidGroup &G, const Word &w)
	Create a normal form of a braid word.
BKLLeftNormalForm &	operator= (const BKLLeftNormalForm &bkl)
	Assignment operator.
BKLLeftNormalForm &	operator= (const NF &bkl)
	Assignment operator.
BKLLeftNormalForm &	operator*= (const BKLLeftNormalForm &bkl)
BKLLeftNormalForm	operator* (const BKLLeftNormalForm &bkl) const
BKLLeftNormalForm	operator- () const
bool	operator== (const BKLLeftNormalForm &bkl) const
bool	operator!= (const BKLLeftNormalForm &bkl) const
bool	operator< (const BKLLeftNormalForm &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.
pair< bool, Word >	areConjugate (const BKLLeftNormalForm &bkl) const
	Check if two normal forms are conjugate.
pair< BKLLeftNormalForm, BKLLeftNormalForm >	cycle () const
	Cycle operation.
pair< BKLLeftNormalForm, BKLLeftNormalForm >	decycle () const
	Decycle operation.
pair< BKLLeftNormalForm, BKLLeftNormalForm >	findSSSRepresentative () const
	Find a representative of the supper summit set.
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 $\delta = (1,2,3,\ldots,n-1,0)$ .
Private Types
enum	transformationResult { TWO_MULTIPLIERS, ONE_MULTIPLIER, NO_CHANGE }
	The result of the transformation. More...
Private Member Functions
NF	inverse () const
	Invert the normal form.
NF	multiply (const BKLLeftNormalForm &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.
static set< Permutation >	getSimpleConjugators (const NF &bkl)
	Function returns a set of permutation braids (well, permutations) such that ... Must be checked.
static set< Permutation >	getSimpleSummitConjugators (const NF &bkl)
	Function returns a set of permutation braids (well, permutations) such that ... Must be checked.
static void	adjustDecomposition (int rank, int &power, list< Permutation > &decomp)
	Function adjusting the decomposition in a normal form.
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

Birman-Ko-Lee left 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 34 of file BKLLeftNormalForm.h.

Member Typedef Documentation

typedef triple< int , int , list< Permutation > > BKLLeftNormalForm::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 45 of file BKLLeftNormalForm.h.

Member Enumeration Documentation

enum BKLLeftNormalForm::transformationResult [private]

The result of the transformation.

Enumerator:

TWO_MULTIPLIERS
ONE_MULTIPLIER
NO_CHANGE

Definition at line 192 of file BKLLeftNormalForm.h.

Constructor & Destructor Documentation

BKLLeftNormalForm::BKLLeftNormalForm ( ) [inline]

Create trivial normal form.

Definition at line 55 of file BKLLeftNormalForm.h.

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

Copy constructor.

Definition at line 60 of file BKLLeftNormalForm.h.

BKLLeftNormalForm::BKLLeftNormalForm	(	int	rank,
		int	p,
		const list< Permutation > &	d
	)			`[inline]`

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

Definition at line 66 of file BKLLeftNormalForm.h.

BKLLeftNormalForm::BKLLeftNormalForm ( const NF & bkl ) [inline]

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

Definition at line 72 of file BKLLeftNormalForm.h.

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

Create a normal form of a braid word.

Member Function Documentation

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

Function adjusting the decomposition in a normal form.

pair< bool , Word > BKLLeftNormalForm::areConjugate ( const BKLLeftNormalForm & bkl ) const

Check if two normal forms are conjugate.

pair< BKLLeftNormalForm , BKLLeftNormalForm > BKLLeftNormalForm::cycle ( ) const

Cycle operation.

pair< BKLLeftNormalForm , BKLLeftNormalForm > BKLLeftNormalForm::decycle ( ) const

Decycle operation.

pair< BKLLeftNormalForm , BKLLeftNormalForm > BKLLeftNormalForm::findSSSRepresentative ( ) const

Find a representative of the supper summit set.

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

Get the decomposition.

Definition at line 126 of file BKLLeftNormalForm.h.

References theDecomposition.

int BKLLeftNormalForm::getPower ( ) const [inline]

Get the power of omega.

Definition at line 124 of file BKLLeftNormalForm.h.

References theOmegaPower.

static set<Permutation> BKLLeftNormalForm::getSimpleConjugators ( const NF & bkl ) [static, private]

Function returns a set of permutation braids (well, permutations) such that ... Must be checked.

static set<Permutation> BKLLeftNormalForm::getSimpleSummitConjugators ( const NF & bkl ) [static, private]

Function returns a set of permutation braids (well, permutations) such that ... Must be checked.

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

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

Definition at line 135 of file BKLLeftNormalForm.h.

Word BKLLeftNormalForm::getWord ( ) const

Return a word represented by the normal form.

NF BKLLeftNormalForm::inverse ( ) const [private]

Invert the normal form.

bool BKLLeftNormalForm::isTrivial ( ) const [inline]

Check if the normal form os trivial.

Definition at line 132 of file BKLLeftNormalForm.h.

References theDecomposition, and theOmegaPower.

NF BKLLeftNormalForm::multiply ( const BKLLeftNormalForm & bkl ) const [private]

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

BKLLeftNormalForm::operator NF ( ) const [inline]

Get the presentation of the normal form.

Definition at line 129 of file BKLLeftNormalForm.h.

References theDecomposition, theOmegaPower, and theRank.

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

BKLLeftNormalForm BKLLeftNormalForm::operator* ( const BKLLeftNormalForm & bkl ) const

BKLLeftNormalForm& BKLLeftNormalForm::operator*= ( const BKLLeftNormalForm & bkl )

BKLLeftNormalForm BKLLeftNormalForm::operator- ( ) const

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

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

Assignment operator.

Definition at line 91 of file BKLLeftNormalForm.h.

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

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

Assignment operator.

Definition at line 83 of file BKLLeftNormalForm.h.

References theDecomposition, theOmegaPower, and theRank.

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

void BKLLeftNormalForm::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 171 of file BKLLeftNormalForm.h.

References theDecomposition.

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

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

Definition at line 167 of file BKLLeftNormalForm.h.

References theOmegaPower.

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

Main function for computing the normal form of the word.

Member Data Documentation

list< Permutation > BKLLeftNormalForm::theDecomposition [private]

Sequence of permutations.

Definition at line 227 of file BKLLeftNormalForm.h.

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

int BKLLeftNormalForm::theOmegaPower [private]

Power of omega.

Definition at line 224 of file BKLLeftNormalForm.h.

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

int BKLLeftNormalForm::theRank [private]

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

Definition at line 221 of file BKLLeftNormalForm.h.

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

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

BraidGroup/include/BKLLeftNormalForm.h

BKLLeftNormalForm Class Reference

Public Types

Public Member Functions

Static Public Member Functions

Private Types

Private Member Functions