BraidWord

class braidgenerator.BraidWord(**kwargs)

A BraidWord represents a mathematical braid. A mathematical braid is a word expressed as a finite list of Artin generators or their inverses. Generators are represented here as positive or negative integers. An object of BraidWord (i.e. a braid) is the main component acted upon by the MarkovChain class.

braidword list: List containing integers (representing Artin generators) to be used as initial word.

word list: word of BraidWord, made up of list containing integer generators.
largestGenerator int: Largest Generator in word.
genCount list: Array of quantity of generators in word.

<BraidWord>.methods

: Boolean stating if cancel at index is a valid move. Cancellation refers to the cancellation of a generator and its consecutive inverse in the braid group.

index int: Index where canCancel is to be checked.

Returns

True if successful, False otherwise.

: Checks if destabilize() is a valid move. Destabilization refers to destabilization of a closed braid. destabilize is the inverse operation of stabilize and is only permissible when the `largestGenerator` appears exactly once at the last index of `word`.

Returns

True if successful, False otherwise.

: Checks if flip can be performed at index. A `flip' refers to the second braid relation in the braid group.

index int: Index where canFlip is to be checked.

Returns

True if successful, False otherwise.

: Checks if transpose at index is a valid move. A transpose is the first braid relation in the braid group.

index int: Index where canTranspose is to be checked.

Returns

True if successful, False otherwise.

: Performs cancellation of the generators at index and (index+1) modulo length and decreases length of word by two. Cancellation refers to the cancellation of a generator and its consecutive inverse in the braid group.

index int: Index where cancel is to be performed.

Returns

Boolean that triggers state of Markov step in logs.

: Method to conjugate the word at index. Conjugation is defined for closed braids by AB = BA.

index int: Index where conjugate is to be performed.

Returns

Boolean that triggers state of Markov step in logs.

: Destabilizes the BraidWord, decreases word length by one, and decreases largestGenerator if successful. Destabilization is a Markov Move defined for closed braids.

Returns

Boolean that triggers state of Markov step in logs.

: Flips generators at index, (index+1) modulo length, and (index+2) modulo length. A flip performs the second braid relation in the braid group.

index int: Index where flip is to be performed.

Returns

Boolean that triggers state of Markov step in logs.

: Insert a generator and its inverse into word. Will insert the given generator and its inverse at index and increases word length by two. Insertion is permissible because the braid group is a group.

index int: Index where insert is to be performed.
generator int: Generator to be inserted into word.

Returns

Boolean that triggers state of Markov step in logs.

: Dynamically returns the length of the word.

Returns

Length of word.

: Stabilizes the BraidWord, increases word length by one, and increases largestGenerator by one if successful. Stabilization is a Markov Move defined for closed braids.

Returns

Boolean that triggers state of Markov step in logs.

: Transposes generators at index and (index+1) modulo length. Transpose performs the first braid relation in the braid group.

index int: Index where transpose is to be performed.

Returns

Boolean that triggers state of Markov step in logs.

: The crossing_change method performs a crossing change on a BraidWord. That is, it returns a new BraidWord whereby the generator at a specified index is inverted. The resulting index can be set randomly with random_index set to True, or it can be set manually by setting random_index=False and assigning index with an integer value between zero and the length of the word of the BraidWord.

random_index bool: Determines whether method randomly selects index or not.
index int: Index where generator will be inverted.

Returns

BraidWord

: The resolution method performs a resolution on a BraidWord. That is, it returns a new BraidWord whereby the generator at a specified index is removed and the length of the new BraidWord is decreased by one. The resulting index can be set randomly with random_index set to True, or it can be set manually by setting random_index=False and assigning index with an integer value between zero and the length of the word of the BraidWord.

random_index bool: Determines whether method randomly selects index or not.
index int: Index where generator will be removed.

Returns

BraidWord