#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 ![]() | |
void | setDecomposition (const list< Permutation > &d) |
Set the decomposition ![]() | |
Static Public Member Functions | |
static Permutation | getTinyTwistPermutation (int theIndex) |
Returns a cyclic permutation ![]() | |
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. |
The standard presentation for BKL is a pair , where
and each
is a permutation.
Definition at line 34 of file BKLLeftNormalForm.h.
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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. |
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The result of the transformation.
Definition at line 192 of file BKLLeftNormalForm.h. |
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Create trivial normal form.
Definition at line 55 of file BKLLeftNormalForm.h. |
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Copy constructor.
Definition at line 60 of file BKLLeftNormalForm.h. |
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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. |
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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. |
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Create a normal form of a braid word.
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Function adjusting the decomposition in a normal form.
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Check if two normal forms are conjugate.
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Cycle operation.
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Decycle operation.
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Find a representative of the supper summit set.
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Get the decomposition.
Definition at line 126 of file BKLLeftNormalForm.h. References theDecomposition. |
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Get the power of omega.
Definition at line 124 of file BKLLeftNormalForm.h. References theOmegaPower. |
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Function returns a set of permutation braids (well, permutations) such that ... Must be checked.
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Function returns a set of permutation braids (well, permutations) such that ... Must be checked.
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Returns a cyclic permutation
Definition at line 135 of file BKLLeftNormalForm.h. |
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Return a word represented by the normal form.
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Invert the normal form.
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Check if the normal form os trivial.
Definition at line 132 of file BKLLeftNormalForm.h. References theDecomposition, and theOmegaPower. |
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Multiply the normal form by another normal form on the right.
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Get the presentation of the normal form.
Definition at line 129 of file BKLLeftNormalForm.h. References theDecomposition, theOmegaPower, and theRank. |
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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. |
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Assignment operator.
Definition at line 83 of file BKLLeftNormalForm.h. References theDecomposition, theOmegaPower, and theRank. |
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Set the decomposition
Definition at line 171 of file BKLLeftNormalForm.h. References theDecomposition. |
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Set the power of
Definition at line 167 of file BKLLeftNormalForm.h. References theOmegaPower. |
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Main function for computing the normal form of the word.
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Sequence of permutations.
Definition at line 227 of file BKLLeftNormalForm.h. Referenced by getDecomposition(), isTrivial(), operator NF(), operator=(), and setDecomposition(). |
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Power of omega.
Definition at line 224 of file BKLLeftNormalForm.h. Referenced by getPower(), isTrivial(), operator NF(), operator=(), and setPower(). |
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The rank of the braid group (number of strands).
Definition at line 221 of file BKLLeftNormalForm.h. Referenced by operator NF(), and operator=(). |