Braid groups¶
Braid groups are implemented as a particular case of finitely presented groups, but with a lot of specific methods for braids.
A braid group can be created by giving the number of strands, and the name of the generators:
sage: BraidGroup(3)
Braid group on 3 strands
sage: BraidGroup(3,'a')
Braid group on 3 strands
sage: BraidGroup(3,'a').gens()
(a0, a1)
sage: BraidGroup(3,'a,b').gens()
(a, b)
>>> from sage.all import *
>>> BraidGroup(Integer(3))
Braid group on 3 strands
>>> BraidGroup(Integer(3),'a')
Braid group on 3 strands
>>> BraidGroup(Integer(3),'a').gens()
(a0, a1)
>>> BraidGroup(Integer(3),'a,b').gens()
(a, b)
The elements can be created by operating with the generators, or by passing a list with the indices of the letters to the group:
sage: B.<s0,s1,s2> = BraidGroup(4)
sage: s0*s1*s0
s0*s1*s0
sage: B([1,2,1])
s0*s1*s0
>>> from sage.all import *
>>> B = BraidGroup(Integer(4), names=('s0', 's1', 's2',)); (s0, s1, s2,) = B._first_ngens(3)
>>> s0*s1*s0
s0*s1*s0
>>> B([Integer(1),Integer(2),Integer(1)])
s0*s1*s0
The mapping class action of the braid group over the free group is
also implemented, see MappingClassGroupAction for an
explanation. This action is left multiplication of a free group
element by a braid:
sage: B.<b0,b1,b2> = BraidGroup()
sage: F.<f0,f1,f2,f3> = FreeGroup()
sage: B.strands() == F.rank() # necessary for the action to be defined
True
sage: f1 * b1
f1*f2*f1^-1
sage: f0 * b1
f0
sage: f1 * b1
f1*f2*f1^-1
sage: f1^-1 * b1
f1*f2^-1*f1^-1
>>> from sage.all import *
>>> B = BraidGroup(names=('b0', 'b1', 'b2',)); (b0, b1, b2,) = B._first_ngens(3)
>>> F = FreeGroup(names=('f0', 'f1', 'f2', 'f3',)); (f0, f1, f2, f3,) = F._first_ngens(4)
>>> B.strands() == F.rank() # necessary for the action to be defined
True
>>> f1 * b1
f1*f2*f1^-1
>>> f0 * b1
f0
>>> f1 * b1
f1*f2*f1^-1
>>> f1**-Integer(1) * b1
f1*f2^-1*f1^-1
AUTHORS:
Miguel Angel Marco Buzunariz
Volker Braun
Søren Fuglede Jørgensen
Robert Lipshitz
Thierry Monteil: add a
__hash__method consistent with the word problem to ensure correct Cayley graph computations.Sebastian Oehms (July and Nov 2018): add other versions for burau_matrix (unitary + simple, see Issue #25760 and Issue #26657)
Moritz Firsching (Sept 2021): Colored Jones polynomial
Sebastian Oehms (May 2022): add
links_gould_polynomial()
- class sage.groups.braid.Braid(parent, x, check=True)[source]¶
Bases:
FiniteTypeArtinGroupElementAn element of a braid group.
It is a particular case of element of a finitely presented group.
EXAMPLES:
sage: B.<s0,s1,s2> = BraidGroup(4) sage: B Braid group on 4 strands sage: s0*s1/s2/s1 s0*s1*s2^-1*s1^-1 sage: B((1, 2, -3, -2)) s0*s1*s2^-1*s1^-1
>>> from sage.all import * >>> B = BraidGroup(Integer(4), names=('s0', 's1', 's2',)); (s0, s1, s2,) = B._first_ngens(3) >>> B Braid group on 4 strands >>> s0*s1/s2/s1 s0*s1*s2^-1*s1^-1 >>> B((Integer(1), Integer(2), -Integer(3), -Integer(2))) s0*s1*s2^-1*s1^-1
- LKB_matrix(variables='x,y')[source]¶
Return the Lawrence-Krammer-Bigelow representation matrix.
The matrix is expressed in the basis \(\{e_{i, j} \mid 1\leq i < j \leq n\}\), where the indices are ordered lexicographically. It is a matrix whose entries are in the ring of Laurent polynomials on the given variables. By default, the variables are
'x'and'y'.INPUT:
variables– string (default:'x,y'); a string containing the names of the variables, separated by a comma
OUTPUT: the matrix corresponding to the Lawrence-Krammer-Bigelow representation of the braid
EXAMPLES:
sage: B = BraidGroup(3) sage: b = B([1, 2, 1]) sage: b.LKB_matrix() [ 0 -x^4*y + x^3*y -x^4*y] [ 0 -x^3*y 0] [ -x^2*y x^3*y - x^2*y 0] sage: c = B([2, 1, 2]) sage: c.LKB_matrix() [ 0 -x^4*y + x^3*y -x^4*y] [ 0 -x^3*y 0] [ -x^2*y x^3*y - x^2*y 0]
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> b = B([Integer(1), Integer(2), Integer(1)]) >>> b.LKB_matrix() [ 0 -x^4*y + x^3*y -x^4*y] [ 0 -x^3*y 0] [ -x^2*y x^3*y - x^2*y 0] >>> c = B([Integer(2), Integer(1), Integer(2)]) >>> c.LKB_matrix() [ 0 -x^4*y + x^3*y -x^4*y] [ 0 -x^3*y 0] [ -x^2*y x^3*y - x^2*y 0]
REFERENCES:
- TL_matrix(drain_size, variab=None, sparse=True)[source]¶
Return the matrix representation of the Temperley–Lieb–Jones representation of the braid in a certain basis.
The basis is given by non-intersecting pairings of \((n+d)\) points, where \(n\) is the number of strands, \(d\) is given by
drain_size, and the pairings satisfy certain rules. SeeTL_basis_with_drain()for details.We use the convention that the eigenvalues of the standard generators are \(1\) and \(-A^4\), where \(A\) is a variable of a Laurent polynomial ring.
When \(d = n - 2\) and the variables are picked appropriately, the resulting representation is equivalent to the reduced Burau representation.
INPUT:
drain_size– integer between 0 and the number of strands (both inclusive)variab– variable (default:None); the variable in the entries of the matrices; ifNone, then use a default variable in \(\ZZ[A,A^{-1}]\)sparse– boolean (default:True); whether or not the result should be given as a sparse matrix
OUTPUT: the matrix of the TL representation of the braid
The parameter
sparsecan be set toFalseif it is expected that the resulting matrix will not be sparse. We currently make no attempt at guessing this.EXAMPLES:
Let us calculate a few examples for \(B_4\) with \(d = 0\):
sage: B = BraidGroup(4) sage: b = B([1, 2, -3]) sage: b.TL_matrix(0) [1 - A^4 -A^-2] [ -A^6 0] sage: R.<x> = LaurentPolynomialRing(GF(2)) sage: b.TL_matrix(0, variab=x) [1 + x^4 x^-2] [ x^6 0] sage: b = B([]) sage: b.TL_matrix(0) [1 0] [0 1]
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b = B([Integer(1), Integer(2), -Integer(3)]) >>> b.TL_matrix(Integer(0)) [1 - A^4 -A^-2] [ -A^6 0] >>> R = LaurentPolynomialRing(GF(Integer(2)), names=('x',)); (x,) = R._first_ngens(1) >>> b.TL_matrix(Integer(0), variab=x) [1 + x^4 x^-2] [ x^6 0] >>> b = B([]) >>> b.TL_matrix(Integer(0)) [1 0] [0 1]
Test of one of the relations in \(B_8\):
sage: B = BraidGroup(8) sage: d = 0 sage: B([4,5,4]).TL_matrix(d) == B([5,4,5]).TL_matrix(d) True
>>> from sage.all import * >>> B = BraidGroup(Integer(8)) >>> d = Integer(0) >>> B([Integer(4),Integer(5),Integer(4)]).TL_matrix(d) == B([Integer(5),Integer(4),Integer(5)]).TL_matrix(d) True
An element of the kernel of the Burau representation, following [Big1999]:
sage: B = BraidGroup(6) sage: psi1 = B([4, -5, -2, 1]) sage: psi2 = B([-4, 5, 5, 2, -1, -1]) sage: w1 = psi1^(-1) * B([3]) * psi1 sage: w2 = psi2^(-1) * B([3]) * psi2 sage: (w1 * w2 * w1^(-1) * w2^(-1)).TL_matrix(4) [1 0 0 0 0] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1]
>>> from sage.all import * >>> B = BraidGroup(Integer(6)) >>> psi1 = B([Integer(4), -Integer(5), -Integer(2), Integer(1)]) >>> psi2 = B([-Integer(4), Integer(5), Integer(5), Integer(2), -Integer(1), -Integer(1)]) >>> w1 = psi1**(-Integer(1)) * B([Integer(3)]) * psi1 >>> w2 = psi2**(-Integer(1)) * B([Integer(3)]) * psi2 >>> (w1 * w2 * w1**(-Integer(1)) * w2**(-Integer(1))).TL_matrix(Integer(4)) [1 0 0 0 0] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1]
REFERENCES:
- alexander_polynomial(var='t', normalized=True)[source]¶
Return the Alexander polynomial of the closure of the braid.
INPUT:
var– string (default:'t'); the name of the variable in the entries of the matrixnormalized– boolean (default:True); whether to return the normalized Alexander polynomial
OUTPUT: the Alexander polynomial of the braid closure of the braid
This is computed using the reduced Burau representation. The unnormalized Alexander polynomial is a Laurent polynomial, which is only well-defined up to multiplication by plus or minus times a power of \(t\).
We normalize the polynomial by dividing by the largest power of \(t\) and then if the resulting constant coefficient is negative, we multiply by \(-1\).
EXAMPLES:
We first construct the trefoil:
sage: B = BraidGroup(3) sage: b = B([1,2,1,2]) sage: b.alexander_polynomial(normalized=False) 1 - t + t^2 sage: b.alexander_polynomial() t^-2 - t^-1 + 1
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> b = B([Integer(1),Integer(2),Integer(1),Integer(2)]) >>> b.alexander_polynomial(normalized=False) 1 - t + t^2 >>> b.alexander_polynomial() t^-2 - t^-1 + 1
Next we construct the figure 8 knot:
sage: b = B([-1,2,-1,2]) sage: b.alexander_polynomial(normalized=False) -t^-2 + 3*t^-1 - 1 sage: b.alexander_polynomial() t^-2 - 3*t^-1 + 1
>>> from sage.all import * >>> b = B([-Integer(1),Integer(2),-Integer(1),Integer(2)]) >>> b.alexander_polynomial(normalized=False) -t^-2 + 3*t^-1 - 1 >>> b.alexander_polynomial() t^-2 - 3*t^-1 + 1
Our last example is the Kinoshita-Terasaka knot:
sage: B = BraidGroup(4) sage: b = B([1,1,1,3,3,2,-3,-1,-1,2,-1,-3,-2]) sage: b.alexander_polynomial(normalized=False) -t^-1 sage: b.alexander_polynomial() 1
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b = B([Integer(1),Integer(1),Integer(1),Integer(3),Integer(3),Integer(2),-Integer(3),-Integer(1),-Integer(1),Integer(2),-Integer(1),-Integer(3),-Integer(2)]) >>> b.alexander_polynomial(normalized=False) -t^-1 >>> b.alexander_polynomial() 1
REFERENCES:
- annular_khovanov_complex(qagrad=None, ring=None)[source]¶
Return the annular Khovanov complex of the closure of a braid, as defined in [BG2013].
INPUT:
qagrad– tuple of quantum and annular grading for which to compute the chain complex; if not specified all gradings are computedring– (default:ZZ) the coefficient ring
OUTPUT:
The annular Khovanov complex of the braid, given as a dictionary whose keys are tuples of quantum and annular grading. If
qagradis specified only return the chain complex of that grading.EXAMPLES:
sage: B = BraidGroup(3) sage: b = B([1,-2,1,-2]) sage: C = b.annular_khovanov_complex() sage: C {(-5, -1): Chain complex with at most 1 nonzero terms over Integer Ring, (-3, -3): Chain complex with at most 1 nonzero terms over Integer Ring, (-3, -1): Chain complex with at most 2 nonzero terms over Integer Ring, (-3, 1): Chain complex with at most 1 nonzero terms over Integer Ring, (-1, -1): Chain complex with at most 5 nonzero terms over Integer Ring, (-1, 1): Chain complex with at most 2 nonzero terms over Integer Ring, (1, -1): Chain complex with at most 2 nonzero terms over Integer Ring, (1, 1): Chain complex with at most 5 nonzero terms over Integer Ring, (3, -1): Chain complex with at most 1 nonzero terms over Integer Ring, (3, 1): Chain complex with at most 2 nonzero terms over Integer Ring, (3, 3): Chain complex with at most 1 nonzero terms over Integer Ring, (5, 1): Chain complex with at most 1 nonzero terms over Integer Ring} sage: C[1,-1].homology() {1: Z x Z, 2: 0}
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> b = B([Integer(1),-Integer(2),Integer(1),-Integer(2)]) >>> C = b.annular_khovanov_complex() >>> C {(-5, -1): Chain complex with at most 1 nonzero terms over Integer Ring, (-3, -3): Chain complex with at most 1 nonzero terms over Integer Ring, (-3, -1): Chain complex with at most 2 nonzero terms over Integer Ring, (-3, 1): Chain complex with at most 1 nonzero terms over Integer Ring, (-1, -1): Chain complex with at most 5 nonzero terms over Integer Ring, (-1, 1): Chain complex with at most 2 nonzero terms over Integer Ring, (1, -1): Chain complex with at most 2 nonzero terms over Integer Ring, (1, 1): Chain complex with at most 5 nonzero terms over Integer Ring, (3, -1): Chain complex with at most 1 nonzero terms over Integer Ring, (3, 1): Chain complex with at most 2 nonzero terms over Integer Ring, (3, 3): Chain complex with at most 1 nonzero terms over Integer Ring, (5, 1): Chain complex with at most 1 nonzero terms over Integer Ring} >>> C[Integer(1),-Integer(1)].homology() {1: Z x Z, 2: 0}
- annular_khovanov_homology(qagrad=None, ring=Integer Ring)[source]¶
Return the annular Khovanov homology of a closure of a braid.
INPUT:
qagrad– (optional) tuple of quantum and annular grading for which to compute the homologyring– (default:ZZ) the coefficient ring
OUTPUT:
If
qagradisNone, return a dictionary of homologies in all gradings indexed by grading. Ifqagradis specified, return the homology of that grading.Note
This is a simple wrapper around
annular_khovanov_complex()to compute homology from it.EXAMPLES:
sage: B = BraidGroup(4) sage: b = B([1,3,-2]) sage: b.annular_khovanov_homology() {(-3, -4): {0: Z}, (-3, -2): {-1: Z}, (-1, -2): {-1: 0, 0: Z x Z x Z, 1: 0}, (-1, 0): {-1: Z x Z}, (1, -2): {1: Z x Z}, (1, 0): {-1: 0, 0: Z x Z x Z x Z, 1: 0, 2: 0}, (1, 2): {-1: Z}, (3, 0): {1: Z x Z x Z, 2: 0}, (3, 2): {-1: 0, 0: Z x Z x Z, 1: 0}, (5, 0): {2: Z}, (5, 2): {1: Z x Z}, (5, 4): {0: Z}} sage: B = BraidGroup(2) sage: b = B([1,1,1]) sage: b.annular_khovanov_homology((7,0)) {2: 0, 3: C2}
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b = B([Integer(1),Integer(3),-Integer(2)]) >>> b.annular_khovanov_homology() {(-3, -4): {0: Z}, (-3, -2): {-1: Z}, (-1, -2): {-1: 0, 0: Z x Z x Z, 1: 0}, (-1, 0): {-1: Z x Z}, (1, -2): {1: Z x Z}, (1, 0): {-1: 0, 0: Z x Z x Z x Z, 1: 0, 2: 0}, (1, 2): {-1: Z}, (3, 0): {1: Z x Z x Z, 2: 0}, (3, 2): {-1: 0, 0: Z x Z x Z, 1: 0}, (5, 0): {2: Z}, (5, 2): {1: Z x Z}, (5, 4): {0: Z}} >>> B = BraidGroup(Integer(2)) >>> b = B([Integer(1),Integer(1),Integer(1)]) >>> b.annular_khovanov_homology((Integer(7),Integer(0))) {2: 0, 3: C2}
- burau_matrix(var='t', reduced=False)[source]¶
Return the Burau matrix of the braid.
INPUT:
var– string (default:'t'); the name of the variable in the entries of the matrixreduced– boolean (default:False); whether to return the reduced or unreduced Burau representation, can be one of the following:Trueor'increasing'– returns the reduced form using the basis given by \(e_1 - e_i\) for \(2 \leq i \leq n\)'unitary'– the unitary form according to Squier [Squ1984]'simple'– returns the reduced form using the basis given by simple roots \(e_i - e_{i+1}\), which yields the matrices given on the Wikipedia page
OUTPUT:
The Burau matrix of the braid. It is a matrix whose entries are Laurent polynomials in the variable
var. IfreducedisTrue, return the matrix for the reduced Burau representation instead in the format specified. Ifreducedis'unitary', a tripleM, Madj, His returned, whereMis the Burau matrix in the unitary form,Madjthe adjoined toMandHthe hermitian form.EXAMPLES:
sage: B = BraidGroup(4) sage: B.inject_variables() Defining s0, s1, s2 sage: b = s0 * s1 / s2 / s1 sage: b.burau_matrix() [ 1 - t 0 t - t^2 t^2] [ 1 0 0 0] [ 0 0 1 0] [ 0 t^-2 -t^-2 + t^-1 -t^-1 + 1] sage: s2.burau_matrix('x') [ 1 0 0 0] [ 0 1 0 0] [ 0 0 1 - x x] [ 0 0 1 0] sage: s0.burau_matrix(reduced=True) [-t 0 0] [-t 1 0] [-t 0 1]
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> B.inject_variables() Defining s0, s1, s2 >>> b = s0 * s1 / s2 / s1 >>> b.burau_matrix() [ 1 - t 0 t - t^2 t^2] [ 1 0 0 0] [ 0 0 1 0] [ 0 t^-2 -t^-2 + t^-1 -t^-1 + 1] >>> s2.burau_matrix('x') [ 1 0 0 0] [ 0 1 0 0] [ 0 0 1 - x x] [ 0 0 1 0] >>> s0.burau_matrix(reduced=True) [-t 0 0] [-t 1 0] [-t 0 1]
Using the different reduced forms:
sage: b.burau_matrix(reduced='simple') [ 1 - t -t^-1 + 1 -1] [ 1 -t^-1 + 1 -1] [ 1 -t^-1 0] sage: M, Madj, H = b.burau_matrix(reduced='unitary') sage: M [-t^-2 + 1 t t^2] [ t^-1 - t 1 - t^2 -t^3] [ -t^-2 -t^-1 0] sage: Madj [ 1 - t^2 -t^-1 + t -t^2] [ t^-1 -t^-2 + 1 -t] [ t^-2 -t^-3 0] sage: H [t^-1 + t -1 0] [ -1 t^-1 + t -1] [ 0 -1 t^-1 + t] sage: M * H * Madj == H True
>>> from sage.all import * >>> b.burau_matrix(reduced='simple') [ 1 - t -t^-1 + 1 -1] [ 1 -t^-1 + 1 -1] [ 1 -t^-1 0] >>> M, Madj, H = b.burau_matrix(reduced='unitary') >>> M [-t^-2 + 1 t t^2] [ t^-1 - t 1 - t^2 -t^3] [ -t^-2 -t^-1 0] >>> Madj [ 1 - t^2 -t^-1 + t -t^2] [ t^-1 -t^-2 + 1 -t] [ t^-2 -t^-3 0] >>> H [t^-1 + t -1 0] [ -1 t^-1 + t -1] [ 0 -1 t^-1 + t] >>> M * H * Madj == H True
The adjoined matrix (
Madjin the above example) matches the output ofsage.groups.artin.ArtinGroupElement.burau_matrix():sage: from sage.groups.artin import ArtinGroupElement sage: Madj == ArtinGroupElement.burau_matrix(b) True sage: a = s0^2 * s1 * s0 * s2 *s1 * ~s0 * s1^3 * s0 * s2 * s1^-2 * s0 sage: a.burau_matrix(reduced='unitary')[1] == ArtinGroupElement.burau_matrix(a) True
>>> from sage.all import * >>> from sage.groups.artin import ArtinGroupElement >>> Madj == ArtinGroupElement.burau_matrix(b) True >>> a = s0**Integer(2) * s1 * s0 * s2 *s1 * ~s0 * s1**Integer(3) * s0 * s2 * s1**-Integer(2) * s0 >>> a.burau_matrix(reduced='unitary')[Integer(1)] == ArtinGroupElement.burau_matrix(a) True
We verify Bigelow’s example that in \(B_5\) the Burau representation is not faithful:
sage: B.<s1,s2,s3,s4> = BraidGroup(5) sage: psi1 = ~s3 * s2 * s1^2 * s2 * s4^3 * s3 * s2 sage: psi2 = ~s4 * s3 * s2 * s1^-2 * s2 * s1^2 * s2^2 * s1 * s4^5 sage: alpha = ~psi1 * s4 * psi1 sage: beta = ~psi2 * s4 * s3 * s2 * s1^2 * s2 * s3 * s4 * psi2 sage: elm = alpha * beta * ~alpha * ~beta sage: elm.burau_matrix() [1 0 0 0 0] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1] sage: elm.burau_matrix(reduced=True) [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] sage: elm.is_one() False
>>> from sage.all import * >>> B = BraidGroup(Integer(5), names=('s1', 's2', 's3', 's4',)); (s1, s2, s3, s4,) = B._first_ngens(4) >>> psi1 = ~s3 * s2 * s1**Integer(2) * s2 * s4**Integer(3) * s3 * s2 >>> psi2 = ~s4 * s3 * s2 * s1**-Integer(2) * s2 * s1**Integer(2) * s2**Integer(2) * s1 * s4**Integer(5) >>> alpha = ~psi1 * s4 * psi1 >>> beta = ~psi2 * s4 * s3 * s2 * s1**Integer(2) * s2 * s3 * s4 * psi2 >>> elm = alpha * beta * ~alpha * ~beta >>> elm.burau_matrix() [1 0 0 0 0] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1] >>> elm.burau_matrix(reduced=True) [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] >>> elm.is_one() False
REFERENCES:
- centralizer()[source]¶
Return a list of generators of the centralizer of the braid.
EXAMPLES:
sage: B = BraidGroup(4) sage: b = B([2, 1, 3, 2]) sage: b.centralizer() [s1*s0*s2*s1, s0*s2]
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b = B([Integer(2), Integer(1), Integer(3), Integer(2)]) >>> b.centralizer() [s1*s0*s2*s1, s0*s2]
- colored_jones_polynomial(N, variab=None, try_inverse=True)[source]¶
Return the colored Jones polynomial of the trace closure of the braid.
INPUT:
N– integer; the number of colorsvariab– (default: \(q\)) the variable in the resulting Laurent polynomialtry_inverse– boolean (default:True); ifTrue, attempt a faster calculation by using the inverse of the braid
ALGORITHM:
The algorithm used is described in [HL2018]. We follow their notation, but work in a suitable free algebra over a Laurent polynomial ring in one variable to simplify bookkeeping.
EXAMPLES:
sage: trefoil = BraidGroup(2)([1,1,1]) sage: trefoil.colored_jones_polynomial(2) q + q^3 - q^4 sage: trefoil.colored_jones_polynomial(4) q^3 + q^7 - q^10 + q^11 - q^13 - q^14 + q^15 - q^17 + q^19 + q^20 - q^21 sage: trefoil.inverse().colored_jones_polynomial(4) -q^-21 + q^-20 + q^-19 - q^-17 + q^-15 - q^-14 - q^-13 + q^-11 - q^-10 + q^-7 + q^-3 sage: figure_eight = BraidGroup(3)([-1, 2, -1, 2]) sage: figure_eight.colored_jones_polynomial(2) q^-2 - q^-1 + 1 - q + q^2 sage: figure_eight.colored_jones_polynomial(3, 'Q') Q^-6 - Q^-5 - Q^-4 + 2*Q^-3 - Q^-2 - Q^-1 + 3 - Q - Q^2 + 2*Q^3 - Q^4 - Q^5 + Q^6
>>> from sage.all import * >>> trefoil = BraidGroup(Integer(2))([Integer(1),Integer(1),Integer(1)]) >>> trefoil.colored_jones_polynomial(Integer(2)) q + q^3 - q^4 >>> trefoil.colored_jones_polynomial(Integer(4)) q^3 + q^7 - q^10 + q^11 - q^13 - q^14 + q^15 - q^17 + q^19 + q^20 - q^21 >>> trefoil.inverse().colored_jones_polynomial(Integer(4)) -q^-21 + q^-20 + q^-19 - q^-17 + q^-15 - q^-14 - q^-13 + q^-11 - q^-10 + q^-7 + q^-3 >>> figure_eight = BraidGroup(Integer(3))([-Integer(1), Integer(2), -Integer(1), Integer(2)]) >>> figure_eight.colored_jones_polynomial(Integer(2)) q^-2 - q^-1 + 1 - q + q^2 >>> figure_eight.colored_jones_polynomial(Integer(3), 'Q') Q^-6 - Q^-5 - Q^-4 + 2*Q^-3 - Q^-2 - Q^-1 + 3 - Q - Q^2 + 2*Q^3 - Q^4 - Q^5 + Q^6
- components_in_closure()[source]¶
Return the number of components of the trace closure of the braid.
OUTPUT: a positive integer
EXAMPLES:
sage: B = BraidGroup(5) sage: b = B([1, -3]) # Three disjoint unknots sage: b.components_in_closure() 3 sage: b = B([1, 2, 3, 4]) # The unknot sage: b.components_in_closure() 1 sage: B = BraidGroup(4) sage: K11n42 = B([1, -2, 3, -2, 3, -2, -2, -1, 2, -3, -3, 2, 2]) sage: K11n42.components_in_closure() 1
>>> from sage.all import * >>> B = BraidGroup(Integer(5)) >>> b = B([Integer(1), -Integer(3)]) # Three disjoint unknots >>> b.components_in_closure() 3 >>> b = B([Integer(1), Integer(2), Integer(3), Integer(4)]) # The unknot >>> b.components_in_closure() 1 >>> B = BraidGroup(Integer(4)) >>> K11n42 = B([Integer(1), -Integer(2), Integer(3), -Integer(2), Integer(3), -Integer(2), -Integer(2), -Integer(1), Integer(2), -Integer(3), -Integer(3), Integer(2), Integer(2)]) >>> K11n42.components_in_closure() 1
- conjugating_braid(other)[source]¶
Return a conjugating braid, if it exists.
INPUT:
other– a braid in the same braid group asself
OUTPUT:
A conjugating braid. More precisely, if the output is \(d\), \(o\) equals
other, and \(s\) equalsselfthen \(o = d^{-1} \cdot s \cdot d\).EXAMPLES:
sage: B = BraidGroup(3) sage: B.one().conjugating_braid(B.one()) 1 sage: B.one().conjugating_braid(B.gen(0)) is None True sage: B.gen(0).conjugating_braid(B.gen(1)) s1*s0 sage: B.gen(0).conjugating_braid(B.gen(1).inverse()) is None True sage: a = B([2, 2, -1, -1]) sage: b = B([2, 1, 2, 1]) sage: c = b * a / b sage: d1 = a.conjugating_braid(c) sage: d1 s1*s0 sage: d1 * c / d1 == a True sage: d1 * a / d1 == c False sage: l = sage.groups.braid.conjugatingbraid(a,c) # needs sage.groups sage: d1 == B._element_from_libbraiding(l) # needs sage.groups True sage: b = B([2, 2, 2, 2, 1]) sage: c = b * a / b sage: d1 = a.conjugating_braid(c) sage: len(d1.Tietze()) 7 sage: d1 * c / d1 == a True sage: d1 * a / d1 == c False sage: d1 s1^2*s0^2*s1^2*s0 sage: l = sage.groups.braid.conjugatingbraid(a,c) # needs sage.groups sage: d2 = B._element_from_libbraiding(l) # needs sage.groups sage: len(d2.Tietze()) # needs sage.groups 13 sage: c.conjugating_braid(b) is None True
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> B.one().conjugating_braid(B.one()) 1 >>> B.one().conjugating_braid(B.gen(Integer(0))) is None True >>> B.gen(Integer(0)).conjugating_braid(B.gen(Integer(1))) s1*s0 >>> B.gen(Integer(0)).conjugating_braid(B.gen(Integer(1)).inverse()) is None True >>> a = B([Integer(2), Integer(2), -Integer(1), -Integer(1)]) >>> b = B([Integer(2), Integer(1), Integer(2), Integer(1)]) >>> c = b * a / b >>> d1 = a.conjugating_braid(c) >>> d1 s1*s0 >>> d1 * c / d1 == a True >>> d1 * a / d1 == c False >>> l = sage.groups.braid.conjugatingbraid(a,c) # needs sage.groups >>> d1 == B._element_from_libbraiding(l) # needs sage.groups True >>> b = B([Integer(2), Integer(2), Integer(2), Integer(2), Integer(1)]) >>> c = b * a / b >>> d1 = a.conjugating_braid(c) >>> len(d1.Tietze()) 7 >>> d1 * c / d1 == a True >>> d1 * a / d1 == c False >>> d1 s1^2*s0^2*s1^2*s0 >>> l = sage.groups.braid.conjugatingbraid(a,c) # needs sage.groups >>> d2 = B._element_from_libbraiding(l) # needs sage.groups >>> len(d2.Tietze()) # needs sage.groups 13 >>> c.conjugating_braid(b) is None True
- cyclic_slidings()[source]¶
Return the braid’s cyclic slidings.
OUTPUT: The braid’s cyclic slidings. Each cyclic sliding is a list of braids.
EXAMPLES:
sage: B = BraidGroup(4) sage: b = B([1, 2, 1, 2, 3, -1, 2, 1]) sage: b.cyclic_slidings() [[s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0^2, s1*s0*s2^2*s1*s0], [s0*s1*s2*s1^2*s0, s0*s1*s2*s1*s0*s2, s1*s0*s2*s0*s1*s2]]
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b = B([Integer(1), Integer(2), Integer(1), Integer(2), Integer(3), -Integer(1), Integer(2), Integer(1)]) >>> b.cyclic_slidings() [[s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0^2, s1*s0*s2^2*s1*s0], [s0*s1*s2*s1^2*s0, s0*s1*s2*s1*s0*s2, s1*s0*s2*s0*s1*s2]]
- deformed_burau_matrix(variab='q')[source]¶
Return the deformed Burau matrix of the braid.
INPUT:
variab– variable (default:q); the variable in the resulting laurent polynomial, which is the base ring for the free algebra constructed
OUTPUT: a matrix with elements in the free algebra
self._algebraEXAMPLES:
sage: B = BraidGroup(4) sage: b = B([1, 2, -3, -2, 3, 1]) sage: db = b.deformed_burau_matrix(); db [ ap_0*ap_5 ... bp_0*ap_1*cm_3*bp_4] ... [ bm_2*bm_3*cp_5 ... bm_2*am_3*bp_4]
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b = B([Integer(1), Integer(2), -Integer(3), -Integer(2), Integer(3), Integer(1)]) >>> db = b.deformed_burau_matrix(); db [ ap_0*ap_5 ... bp_0*ap_1*cm_3*bp_4] ... [ bm_2*bm_3*cp_5 ... bm_2*am_3*bp_4]
We check how this relates to the nondeformed Burau matrix:
sage: def subs_gen(gen, q): ....: gen_str = str(gen) ....: v = q if 'p' in gen_str else 1/q ....: if 'b' in gen_str: ....: return v ....: elif 'a' in gen_str: ....: return 1 - v ....: else: ....: return 1 sage: db_base = db.parent().base_ring() sage: q = db_base.base_ring().gen() sage: db_simp = db.subs({gen: subs_gen(gen, q) ....: for gen in db_base.gens()}) sage: db_simp [ (1-2*q+q^2) (q-q^2) (q-q^2+q^3) (q^2-q^3)] [ (1-q) q 0 0] [ 0 0 (1-q) q] [ (q^-2) 0 -(q^-2-q^-1) -(q^-1-1)] sage: burau = b.burau_matrix(); burau [1 - 2*t + t^2 t - t^2 t - t^2 + t^3 t^2 - t^3] [ 1 - t t 0 0] [ 0 0 1 - t t] [ t^-2 0 -t^-2 + t^-1 -t^-1 + 1] sage: t = burau.parent().base_ring().gen() sage: burau.subs({t:q}).change_ring(db_base) == db_simp True
>>> from sage.all import * >>> def subs_gen(gen, q): ... gen_str = str(gen) ... v = q if 'p' in gen_str else Integer(1)/q ... if 'b' in gen_str: ... return v ... elif 'a' in gen_str: ... return Integer(1) - v ... else: ... return Integer(1) >>> db_base = db.parent().base_ring() >>> q = db_base.base_ring().gen() >>> db_simp = db.subs({gen: subs_gen(gen, q) ... for gen in db_base.gens()}) >>> db_simp [ (1-2*q+q^2) (q-q^2) (q-q^2+q^3) (q^2-q^3)] [ (1-q) q 0 0] [ 0 0 (1-q) q] [ (q^-2) 0 -(q^-2-q^-1) -(q^-1-1)] >>> burau = b.burau_matrix(); burau [1 - 2*t + t^2 t - t^2 t - t^2 + t^3 t^2 - t^3] [ 1 - t t 0 0] [ 0 0 1 - t t] [ t^-2 0 -t^-2 + t^-1 -t^-1 + 1] >>> t = burau.parent().base_ring().gen() >>> burau.subs({t:q}).change_ring(db_base) == db_simp True
- gcd(other)[source]¶
Return the greatest common divisor of the two braids.
INPUT:
other– the other braid with respect with the gcd is computed
EXAMPLES:
sage: B = BraidGroup(3) sage: b = B([1, 2, -1, -2, -2, 1]) sage: c = B([1, 2, 1]) sage: b.gcd(c) s0^-1*s1^-1*s0^-2*s1^2*s0 sage: c.gcd(b) s0^-1*s1^-1*s0^-2*s1^2*s0
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> b = B([Integer(1), Integer(2), -Integer(1), -Integer(2), -Integer(2), Integer(1)]) >>> c = B([Integer(1), Integer(2), Integer(1)]) >>> b.gcd(c) s0^-1*s1^-1*s0^-2*s1^2*s0 >>> c.gcd(b) s0^-1*s1^-1*s0^-2*s1^2*s0
- is_conjugated(other)[source]¶
Check if the two braids are conjugated.
INPUT:
other– the other braid to check for conjugacy
EXAMPLES:
sage: B = BraidGroup(3) sage: a = B([2, 2, -1, -1]) sage: b = B([2, 1, 2, 1]) sage: c = b * a / b sage: c.is_conjugated(a) True sage: c.is_conjugated(b) False
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> a = B([Integer(2), Integer(2), -Integer(1), -Integer(1)]) >>> b = B([Integer(2), Integer(1), Integer(2), Integer(1)]) >>> c = b * a / b >>> c.is_conjugated(a) True >>> c.is_conjugated(b) False
- is_periodic()[source]¶
Check whether the braid is periodic.
EXAMPLES:
sage: B = BraidGroup(3) sage: a = B([2, 2, -1, -1, 2, 2]) sage: b = B([2, 1, 2, 1]) sage: a.is_periodic() False sage: b.is_periodic() True
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> a = B([Integer(2), Integer(2), -Integer(1), -Integer(1), Integer(2), Integer(2)]) >>> b = B([Integer(2), Integer(1), Integer(2), Integer(1)]) >>> a.is_periodic() False >>> b.is_periodic() True
- is_pseudoanosov()[source]¶
Check if the braid is pseudo-Anosov.
EXAMPLES:
sage: B = BraidGroup(3) sage: a = B([2, 2, -1, -1, 2, 2]) sage: b = B([2, 1, 2, 1]) sage: a.is_pseudoanosov() True sage: b.is_pseudoanosov() False
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> a = B([Integer(2), Integer(2), -Integer(1), -Integer(1), Integer(2), Integer(2)]) >>> b = B([Integer(2), Integer(1), Integer(2), Integer(1)]) >>> a.is_pseudoanosov() True >>> b.is_pseudoanosov() False
- is_reducible()[source]¶
Check whether the braid is reducible.
EXAMPLES:
sage: B = BraidGroup(3) sage: b = B([1, 2, -1]) sage: b.is_reducible() True sage: a = B([2, 2, -1, -1, 2, 2]) sage: a.is_reducible() False
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> b = B([Integer(1), Integer(2), -Integer(1)]) >>> b.is_reducible() True >>> a = B([Integer(2), Integer(2), -Integer(1), -Integer(1), Integer(2), Integer(2)]) >>> a.is_reducible() False
- jones_polynomial(variab=None, skein_normalization=False)[source]¶
Return the Jones polynomial of the trace closure of the braid.
The normalization is so that the unknot has Jones polynomial \(1\). If
skein_normalizationisTrue, the variable of the result is replaced by a itself to the power of \(4\), so that the result agrees with the conventions of [Lic1997] (which in particular differs slightly from the conventions used otherwise in this class), had one used the conventional Kauffman bracket variable notation directly.If
variabisNonereturn a polynomial in the variable \(A\) or \(t\), depending on the valueskein_normalization. In particular, ifskein_normalizationisFalse, return the result in terms of the variable \(t\), also used in [Lic1997].INPUT:
variab– variable (default:None); the variable in the resulting polynomial; if unspecified, use either a default variable in \(\ZZ[A,A^{-1}]\) or the variable \(t\) in the symbolic ringskein_normalization– boolean (default:False); determines the variable of the resulting polynomial
OUTPUT:
If
skein_normalizationifFalse, this returns an element in the symbolic ring as the Jones polynomial of the closure might have fractional powers when the closure of the braid is not a knot. Otherwise the result is a Laurent polynomial invariab.EXAMPLES:
The unknot:
sage: B = BraidGroup(9) sage: b = B([1, 2, 3, 4, 5, 6, 7, 8]) sage: b.jones_polynomial() # needs sage.symbolic 1
>>> from sage.all import * >>> B = BraidGroup(Integer(9)) >>> b = B([Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6), Integer(7), Integer(8)]) >>> b.jones_polynomial() # needs sage.symbolic 1
Two unlinked unknots:
sage: B = BraidGroup(2) sage: b = B([]) sage: b.jones_polynomial() # needs sage.symbolic -sqrt(t) - 1/sqrt(t)
>>> from sage.all import * >>> B = BraidGroup(Integer(2)) >>> b = B([]) >>> b.jones_polynomial() # needs sage.symbolic -sqrt(t) - 1/sqrt(t)
The Hopf link:
sage: B = BraidGroup(2) sage: b = B([-1,-1]) sage: b.jones_polynomial() # needs sage.symbolic -1/sqrt(t) - 1/t^(5/2)
>>> from sage.all import * >>> B = BraidGroup(Integer(2)) >>> b = B([-Integer(1),-Integer(1)]) >>> b.jones_polynomial() # needs sage.symbolic -1/sqrt(t) - 1/t^(5/2)
Different representations of the trefoil and one of its mirror:
sage: # needs sage.symbolic sage: B = BraidGroup(2) sage: b = B([-1, -1, -1]) sage: b.jones_polynomial(skein_normalization=True) -A^-16 + A^-12 + A^-4 sage: b.jones_polynomial() 1/t + 1/t^3 - 1/t^4 sage: B = BraidGroup(3) sage: b = B([-1, -2, -1, -2]) sage: b.jones_polynomial(skein_normalization=True) -A^-16 + A^-12 + A^-4 sage: R.<x> = LaurentPolynomialRing(GF(2)) sage: b.jones_polynomial(skein_normalization=True, variab=x) x^-16 + x^-12 + x^-4 sage: B = BraidGroup(3) sage: b = B([1, 2, 1, 2]) sage: b.jones_polynomial(skein_normalization=True) A^4 + A^12 - A^16
>>> from sage.all import * >>> # needs sage.symbolic >>> B = BraidGroup(Integer(2)) >>> b = B([-Integer(1), -Integer(1), -Integer(1)]) >>> b.jones_polynomial(skein_normalization=True) -A^-16 + A^-12 + A^-4 >>> b.jones_polynomial() 1/t + 1/t^3 - 1/t^4 >>> B = BraidGroup(Integer(3)) >>> b = B([-Integer(1), -Integer(2), -Integer(1), -Integer(2)]) >>> b.jones_polynomial(skein_normalization=True) -A^-16 + A^-12 + A^-4 >>> R = LaurentPolynomialRing(GF(Integer(2)), names=('x',)); (x,) = R._first_ngens(1) >>> b.jones_polynomial(skein_normalization=True, variab=x) x^-16 + x^-12 + x^-4 >>> B = BraidGroup(Integer(3)) >>> b = B([Integer(1), Integer(2), Integer(1), Integer(2)]) >>> b.jones_polynomial(skein_normalization=True) A^4 + A^12 - A^16
K11n42 (the mirror of the “Kinoshita-Terasaka” knot) and K11n34 (the mirror of the “Conway” knot):
sage: B = BraidGroup(4) sage: b11n42 = B([1, -2, 3, -2, 3, -2, -2, -1, 2, -3, -3, 2, 2]) sage: b11n34 = B([1, 1, 2, -3, 2, -3, 1, -2, -2, -3, -3]) sage: bool(b11n42.jones_polynomial() == b11n34.jones_polynomial()) # needs sage.symbolic True
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b11n42 = B([Integer(1), -Integer(2), Integer(3), -Integer(2), Integer(3), -Integer(2), -Integer(2), -Integer(1), Integer(2), -Integer(3), -Integer(3), Integer(2), Integer(2)]) >>> b11n34 = B([Integer(1), Integer(1), Integer(2), -Integer(3), Integer(2), -Integer(3), Integer(1), -Integer(2), -Integer(2), -Integer(3), -Integer(3)]) >>> bool(b11n42.jones_polynomial() == b11n34.jones_polynomial()) # needs sage.symbolic True
- lcm(other)[source]¶
Return the least common multiple of the two braids.
INPUT:
other– the other braid with respect with the lcm is computed
EXAMPLES:
sage: B = BraidGroup(3) sage: b = B([1, 2, -1, -2, -2, 1]) sage: c = B([1, 2, 1]) sage: b.lcm(c) (s0*s1)^2*s0
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> b = B([Integer(1), Integer(2), -Integer(1), -Integer(2), -Integer(2), Integer(1)]) >>> c = B([Integer(1), Integer(2), Integer(1)]) >>> b.lcm(c) (s0*s1)^2*s0
- left_normal_form(algorithm='libbraiding')[source]¶
Return the left normal form of the braid.
INPUT:
algorithm– string (default:'artin'); must be one of the following:'artin'– the general method for Artin groups is used'libbraiding'– the algorithm from thelibbraidingpackage
OUTPUT:
A tuple of simple generators in the left normal form. The first element is a power of \(\Delta\), and the rest are elements of the natural section lift from the corresponding symmetric group.
EXAMPLES:
sage: B = BraidGroup(6) sage: B.one().left_normal_form() (1,) sage: b = B([-2, 2, -4, -4, 4, -5, -1, 4, -1, 1]) sage: L1 = b.left_normal_form(); L1 (s0^-1*s1^-1*s0^-1*s2^-1*s1^-1*s0^-1*s3^-1*s2^-1*s1^-1*s0^-1*s4^-1*s3^-1*s2^-1*s1^-1*s0^-1, s0*s2*s1*s0*s3*s2*s1*s0*s4*s3*s2*s1, s3) sage: L1 == b.left_normal_form() True sage: B([1]).left_normal_form(algorithm='artin') (1, s0) sage: B([-3]).left_normal_form(algorithm='artin') (s0^-1*s1^-1*s0^-1*s2^-1*s1^-1*s0^-1*s3^-1*s2^-1*s1^-1*s0^-1*s4^-1*s3^-1*s2^-1*s1^-1*s0^-1, s0*s1*s2*s3*s4*s0*s1*s2*s3*s1*s2*s0*s1*s0) sage: B = BraidGroup(3) sage: B([1,2,-1]).left_normal_form() (s0^-1*s1^-1*s0^-1, s1*s0, s0*s1) sage: B([1,2,1]).left_normal_form() (s0*s1*s0,)
>>> from sage.all import * >>> B = BraidGroup(Integer(6)) >>> B.one().left_normal_form() (1,) >>> b = B([-Integer(2), Integer(2), -Integer(4), -Integer(4), Integer(4), -Integer(5), -Integer(1), Integer(4), -Integer(1), Integer(1)]) >>> L1 = b.left_normal_form(); L1 (s0^-1*s1^-1*s0^-1*s2^-1*s1^-1*s0^-1*s3^-1*s2^-1*s1^-1*s0^-1*s4^-1*s3^-1*s2^-1*s1^-1*s0^-1, s0*s2*s1*s0*s3*s2*s1*s0*s4*s3*s2*s1, s3) >>> L1 == b.left_normal_form() True >>> B([Integer(1)]).left_normal_form(algorithm='artin') (1, s0) >>> B([-Integer(3)]).left_normal_form(algorithm='artin') (s0^-1*s1^-1*s0^-1*s2^-1*s1^-1*s0^-1*s3^-1*s2^-1*s1^-1*s0^-1*s4^-1*s3^-1*s2^-1*s1^-1*s0^-1, s0*s1*s2*s3*s4*s0*s1*s2*s3*s1*s2*s0*s1*s0) >>> B = BraidGroup(Integer(3)) >>> B([Integer(1),Integer(2),-Integer(1)]).left_normal_form() (s0^-1*s1^-1*s0^-1, s1*s0, s0*s1) >>> B([Integer(1),Integer(2),Integer(1)]).left_normal_form() (s0*s1*s0,)
- links_gould_matrix(symbolics=False)[source]¶
Return the representation matrix of
selfaccording to the R-matrix representation being attached to the quantum superalgebra \(\mathfrak{sl}_q(2|1)\).See [MW2012], section 3 and references given there.
INPUT:
symbolics– boolean (default:False); if set toTruethe coefficients will be contained in the symbolic ring. Per default they are elements of a quotient ring of a three variate Laurent polynomial ring.
OUTPUT: the representation matrix of
selfover the ring according to the choice of the keywordsymbolics(see the corresponding explanation)EXAMPLES:
sage: Hopf = BraidGroup(2)([-1, -1]) sage: HopfLG = Hopf.links_gould_matrix() sage: HopfLG.dimensions() (16, 16) sage: HopfLG.base_ring() Univariate Quotient Polynomial Ring in Yrbar over Multivariate Laurent Polynomial Ring in s0r, s1r over Integer Ring with modulus Yr^2 + s0r^2*s1r^2 - s0r^2 - s1r^2 + 1 sage: HopfLGs = Hopf.links_gould_matrix(symbolics=True) # needs sage.symbolic sage: HopfLGs.base_ring() # needs sage.symbolic Symbolic Ring
>>> from sage.all import * >>> Hopf = BraidGroup(Integer(2))([-Integer(1), -Integer(1)]) >>> HopfLG = Hopf.links_gould_matrix() >>> HopfLG.dimensions() (16, 16) >>> HopfLG.base_ring() Univariate Quotient Polynomial Ring in Yrbar over Multivariate Laurent Polynomial Ring in s0r, s1r over Integer Ring with modulus Yr^2 + s0r^2*s1r^2 - s0r^2 - s1r^2 + 1 >>> HopfLGs = Hopf.links_gould_matrix(symbolics=True) # needs sage.symbolic >>> HopfLGs.base_ring() # needs sage.symbolic Symbolic Ring
- links_gould_polynomial(varnames=None, use_symbolics=False)[source]¶
Return the Links-Gould polynomial of the closure of
self.See [MW2012], section 3 and references given there.
INPUT:
varnames– string (default:'t0, t1')
OUTPUT: a Laurent polynomial in the given variable names
EXAMPLES:
sage: Hopf = BraidGroup(2)([-1, -1]) sage: Hopf.links_gould_polynomial() -1 + t1^-1 + t0^-1 - t0^-1*t1^-1 sage: _ == Hopf.links_gould_polynomial(use_symbolics=True) True sage: Hopf.links_gould_polynomial(varnames='a, b') -1 + b^-1 + a^-1 - a^-1*b^-1 sage: _ == Hopf.links_gould_polynomial(varnames='a, b', use_symbolics=True) True
>>> from sage.all import * >>> Hopf = BraidGroup(Integer(2))([-Integer(1), -Integer(1)]) >>> Hopf.links_gould_polynomial() -1 + t1^-1 + t0^-1 - t0^-1*t1^-1 >>> _ == Hopf.links_gould_polynomial(use_symbolics=True) True >>> Hopf.links_gould_polynomial(varnames='a, b') -1 + b^-1 + a^-1 - a^-1*b^-1 >>> _ == Hopf.links_gould_polynomial(varnames='a, b', use_symbolics=True) True
REFERENCES:
- markov_trace(variab=None, normalized=True)[source]¶
Return the Markov trace of the braid.
The normalization is so that in the underlying braid group representation, the eigenvalues of the standard generators of the braid group are \(1\) and \(-A^4\).
INPUT:
variab– variable (default:None); the variable in the resulting polynomial; ifNone, then use the variable \(A\) in \(\ZZ[A,A^{-1}]\)normalized– boolean (default:True); if specified to beFalse, return instead a rescaled Laurent polynomial version of the Markov trace
OUTPUT:
If
normalizedisFalse, return instead the Markov trace of the braid, normalized by a factor of \((A^2+A^{-2})^n\). The result is then a Laurent polynomial invariab. Otherwise it is a quotient of Laurent polynomials invariab.EXAMPLES:
sage: B = BraidGroup(4) sage: b = B([1, 2, -3]) sage: mt = b.markov_trace(); mt A^4/(A^12 + 3*A^8 + 3*A^4 + 1) sage: mt.factor() A^4 * (A^4 + 1)^-3
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b = B([Integer(1), Integer(2), -Integer(3)]) >>> mt = b.markov_trace(); mt A^4/(A^12 + 3*A^8 + 3*A^4 + 1) >>> mt.factor() A^4 * (A^4 + 1)^-3
We now give the non-normalized Markov trace:
sage: mt = b.markov_trace(normalized=False); mt A^-4 + 1 sage: mt.parent() Univariate Laurent Polynomial Ring in A over Integer Ring
>>> from sage.all import * >>> mt = b.markov_trace(normalized=False); mt A^-4 + 1 >>> mt.parent() Univariate Laurent Polynomial Ring in A over Integer Ring
- mirror_image()[source]¶
Return the image of
selfunder the mirror involution (seeBraidGroup_class.mirror_involution()). The link closure of it is mirrored to the closure ofself(see the example below of a positive amphicheiral knot).EXAMPLES:
sage: B5 = BraidGroup(5) sage: b = B5((-1, 2, -3, -1, -3, 4, 2, -3, 2, 4, 2, -3)) # closure K12a_427 sage: bm = b.mirror_image(); bm s0*s1^-1*s2*s0*s2*s3^-1*s1^-1*s2*s1^-1*s3^-1*s1^-1*s2 sage: bm.is_conjugated(b) True sage: bm.is_conjugated(~b) False
>>> from sage.all import * >>> B5 = BraidGroup(Integer(5)) >>> b = B5((-Integer(1), Integer(2), -Integer(3), -Integer(1), -Integer(3), Integer(4), Integer(2), -Integer(3), Integer(2), Integer(4), Integer(2), -Integer(3))) # closure K12a_427 >>> bm = b.mirror_image(); bm s0*s1^-1*s2*s0*s2*s3^-1*s1^-1*s2*s1^-1*s3^-1*s1^-1*s2 >>> bm.is_conjugated(b) True >>> bm.is_conjugated(~b) False
- permutation(W=None)[source]¶
Return the permutation induced by the braid in its strands.
INPUT:
W– (optional) the permutation group to projectselfto; the default isself.parent().coxeter_group()
OUTPUT: the image of
selfunder the natural projection map toWEXAMPLES:
sage: B.<s0,s1,s2> = BraidGroup() sage: S = SymmetricGroup(4) sage: b = s0*s1/s2/s1 sage: c0 = b.permutation(W=S); c0 (1,4,2) sage: c1 = b.permutation(W=Permutations(4)); c1 [4, 1, 3, 2] sage: c1 == b.permutation() True
>>> from sage.all import * >>> B = BraidGroup(names=('s0', 's1', 's2',)); (s0, s1, s2,) = B._first_ngens(3) >>> S = SymmetricGroup(Integer(4)) >>> b = s0*s1/s2/s1 >>> c0 = b.permutation(W=S); c0 (1,4,2) >>> c1 = b.permutation(W=Permutations(Integer(4))); c1 [4, 1, 3, 2] >>> c1 == b.permutation() True
The canonical section from the symmetric group to the braid group (sending a permutation to its associated permutation braid) can be recovered:
sage: B(c0) s0*s1*s2*s1 sage: B(c0) == B(c1) True
>>> from sage.all import * >>> B(c0) s0*s1*s2*s1 >>> B(c0) == B(c1) True
- plot(color='rainbow', orientation='bottom-top', gap=0.05, aspect_ratio=1, axes=False, **kwds)[source]¶
Plot the braid.
The following options are available:
color– (default:'rainbow') the color of the strands. Possible values are:'rainbow', usesrainbow()according to the number of strands.a valid color name for
bezier_path()andline(). Used for all strands.a list or a tuple of colors for each individual strand.
orientation– (default:'bottom-top') determines how the braid is printed. The possible values are:'bottom-top', the braid is printed from bottom to top'top-bottom', the braid is printed from top to bottom'left-right', the braid is printed from left to right
gap– floating point number (default: 0.05); determines the size of the gap left when a strand goes under anotheraspect_ratio– floating point number (default:1); the aspect ratio**kwds– other keyword options that are passed tobezier_path()andline()
EXAMPLES:
sage: B = BraidGroup(3) sage: a = B([2, 2, -1, -1]) sage: b = B([2, 1, 2, 1]) sage: c = b * a / b sage: d = a.conjugating_braid(c) sage: d * c / d == a True sage: d s1*s0 sage: d * a / d == c False sage: B = BraidGroup(4, 's') sage: b = B([1, 2, 3, 1, 2, 1]) sage: b.plot() # needs sage.plot Graphics object consisting of 30 graphics primitives sage: b.plot(color=["red", "blue", "red", "blue"]) # needs sage.plot Graphics object consisting of 30 graphics primitives sage: B.<s,t> = BraidGroup(3) sage: b = t^-1*s^2 sage: b.plot(orientation='left-right', color='red') # needs sage.plot Graphics object consisting of 12 graphics primitives
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> a = B([Integer(2), Integer(2), -Integer(1), -Integer(1)]) >>> b = B([Integer(2), Integer(1), Integer(2), Integer(1)]) >>> c = b * a / b >>> d = a.conjugating_braid(c) >>> d * c / d == a True >>> d s1*s0 >>> d * a / d == c False >>> B = BraidGroup(Integer(4), 's') >>> b = B([Integer(1), Integer(2), Integer(3), Integer(1), Integer(2), Integer(1)]) >>> b.plot() # needs sage.plot Graphics object consisting of 30 graphics primitives >>> b.plot(color=["red", "blue", "red", "blue"]) # needs sage.plot Graphics object consisting of 30 graphics primitives >>> B = BraidGroup(Integer(3), names=('s', 't',)); (s, t,) = B._first_ngens(2) >>> b = t**-Integer(1)*s**Integer(2) >>> b.plot(orientation='left-right', color='red') # needs sage.plot Graphics object consisting of 12 graphics primitives
- plot3d(color='rainbow')[source]¶
Plot the braid in 3d.
The following option is available:
color– (default:'rainbow') the color of the strands. Possible values are:'rainbow', usesrainbow()according to the number of strands.a valid color name for
bezier3d(). Used for all strands.a list or a tuple of colors for each individual strand.
EXAMPLES:
sage: B = BraidGroup(4, 's') sage: b = B([1, 2, 3, 1, 2, 1]) sage: b.plot3d() # needs sage.plot sage.symbolic Graphics3d Object sage: b.plot3d(color='red') # needs sage.plot sage.symbolic Graphics3d Object sage: b.plot3d(color=["red", "blue", "red", "blue"]) # needs sage.plot sage.symbolic Graphics3d Object
>>> from sage.all import * >>> B = BraidGroup(Integer(4), 's') >>> b = B([Integer(1), Integer(2), Integer(3), Integer(1), Integer(2), Integer(1)]) >>> b.plot3d() # needs sage.plot sage.symbolic Graphics3d Object >>> b.plot3d(color='red') # needs sage.plot sage.symbolic Graphics3d Object >>> b.plot3d(color=["red", "blue", "red", "blue"]) # needs sage.plot sage.symbolic Graphics3d Object
- pure_conjugating_braid(other)[source]¶
Return a pure conjugating braid, i.e. a conjugating braid whose associated permutation is the identity, if it exists.
INPUT:
other– a braid in the same braid group asself
OUTPUT:
A pure conjugating braid. More precisely, if the output is \(d\), \(o\) equals
other, and \(s\) equalsselfthen \(o = d^{-1} \cdot s \cdot d\).EXAMPLES:
sage: B = BraidGroup(4) sage: B.one().pure_conjugating_braid(B.one()) 1 sage: B.one().pure_conjugating_braid(B.gen(0)) is None True sage: B.gen(0).pure_conjugating_braid(B.gen(1)) is None True sage: B.gen(0).conjugating_braid(B.gen(2).inverse()) is None True sage: a = B([1, 2, 3]) sage: b = B([3, 2,]) sage: c = b ^ 12 * a / b ^ 12 sage: d1 = a.conjugating_braid(c) sage: len(d1.Tietze()) 30 sage: S = SymmetricGroup(4) sage: d1.permutation(W=S) (1,3)(2,4) sage: d1 * c / d1 == a True sage: d1 * a / d1 == c False sage: d2 = a.pure_conjugating_braid(c) sage: len(d2.Tietze()) 24 sage: d2.permutation(W=S) () sage: d2 * c / d2 == a True sage: d2 (s0*s1*s2^2*s1*s0)^4 sage: a.conjugating_braid(b) is None True sage: a.pure_conjugating_braid(b) is None True sage: a1 = B([1]) sage: a2 = B([2]) sage: a1.conjugating_braid(a2) s1*s0 sage: a1.permutation(W=S) (1,2) sage: a2.permutation(W=S) (2,3) sage: a1.pure_conjugating_braid(a2) is None True sage: (a1^2).conjugating_braid(a2^2) s1*s0 sage: (a1^2).pure_conjugating_braid(a2^2) is None True
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> B.one().pure_conjugating_braid(B.one()) 1 >>> B.one().pure_conjugating_braid(B.gen(Integer(0))) is None True >>> B.gen(Integer(0)).pure_conjugating_braid(B.gen(Integer(1))) is None True >>> B.gen(Integer(0)).conjugating_braid(B.gen(Integer(2)).inverse()) is None True >>> a = B([Integer(1), Integer(2), Integer(3)]) >>> b = B([Integer(3), Integer(2),]) >>> c = b ** Integer(12) * a / b ** Integer(12) >>> d1 = a.conjugating_braid(c) >>> len(d1.Tietze()) 30 >>> S = SymmetricGroup(Integer(4)) >>> d1.permutation(W=S) (1,3)(2,4) >>> d1 * c / d1 == a True >>> d1 * a / d1 == c False >>> d2 = a.pure_conjugating_braid(c) >>> len(d2.Tietze()) 24 >>> d2.permutation(W=S) () >>> d2 * c / d2 == a True >>> d2 (s0*s1*s2^2*s1*s0)^4 >>> a.conjugating_braid(b) is None True >>> a.pure_conjugating_braid(b) is None True >>> a1 = B([Integer(1)]) >>> a2 = B([Integer(2)]) >>> a1.conjugating_braid(a2) s1*s0 >>> a1.permutation(W=S) (1,2) >>> a2.permutation(W=S) (2,3) >>> a1.pure_conjugating_braid(a2) is None True >>> (a1**Integer(2)).conjugating_braid(a2**Integer(2)) s1*s0 >>> (a1**Integer(2)).pure_conjugating_braid(a2**Integer(2)) is None True
- reverse()[source]¶
Return the reverse of
selfobtained by reversing the order of the generators in its word. This defines an anti-involution on the braid group. The link closure of it has the reversed orientation (see the example below of a non reversible knot).EXAMPLES:
sage: b = BraidGroup(3)((1, 1, -2, 1, -2, 1, -2, -2)) # closure K8_17 sage: br = b.reverse(); br s1^-1*(s1^-1*s0)^3*s0 sage: br.is_conjugated(b) False
>>> from sage.all import * >>> b = BraidGroup(Integer(3))((Integer(1), Integer(1), -Integer(2), Integer(1), -Integer(2), Integer(1), -Integer(2), -Integer(2))) # closure K8_17 >>> br = b.reverse(); br s1^-1*(s1^-1*s0)^3*s0 >>> br.is_conjugated(b) False
- right_normal_form()[source]¶
Return the right normal form of the braid.
A tuple of simple generators in the right normal form. The last element is a power of \(\Delta\), and the rest are elements of the natural section lift from the corresponding symmetric group.
EXAMPLES:
sage: B = BraidGroup(4) sage: b = B([1, 2, 1, -2, 3, 1]) sage: b.right_normal_form() (s1*s0, s0*s2, 1)
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b = B([Integer(1), Integer(2), Integer(1), -Integer(2), Integer(3), Integer(1)]) >>> b.right_normal_form() (s1*s0, s0*s2, 1)
- rigidity()[source]¶
Return the rigidity of
self.EXAMPLES:
sage: B = BraidGroup(3) sage: b = B([2, 1, 2, 1]) sage: a = B([2, 2, -1, -1, 2, 2]) sage: a.rigidity() 6 sage: b.rigidity() 0
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> b = B([Integer(2), Integer(1), Integer(2), Integer(1)]) >>> a = B([Integer(2), Integer(2), -Integer(1), -Integer(1), Integer(2), Integer(2)]) >>> a.rigidity() 6 >>> b.rigidity() 0
- sliding_circuits()[source]¶
Return the sliding circuits of the braid.
OUTPUT: list of sliding circuits. Each sliding circuit is itself a list of braids.
EXAMPLES:
sage: B = BraidGroup(3) sage: a = B([2, 2, -1, -1, 2, 2]) sage: a.sliding_circuits() [[(s0^-1*s1^-1*s0^-1)^2*s1^3*s0^2*s1^3], [s0^-1*s1^-1*s0^-2*s1^-1*s0^2*s1^2*s0^3], [s0^-1*s1^-1*s0^-2*s1^-1*s0^3*s1^2*s0^2], [(s0^-1*s1^-1*s0^-1)^2*s1^4*s0^2*s1^2], [(s0^-1*s1^-1*s0^-1)^2*s1^2*s0^2*s1^4], [s0^-1*s1^-1*s0^-2*s1^-1*s0*s1^2*s0^4], [(s0^-1*s1^-1*s0^-1)^2*s1^5*s0^2*s1], [s0^-1*s1^-1*s0^-2*s1^-1*s0^4*s1^2*s0], [(s0^-1*s1^-1*s0^-1)^2*s1*s0^2*s1^5], [s0^-1*s1^-1*s0^-2*s1*s0^5], [(s0^-1*s1^-1*s0^-1)^2*s1*s0^6*s1], [s0^-1*s1^-1*s0^-2*s1^5*s0]] sage: b = B([2, 1, 2, 1]) sage: b.sliding_circuits() [[s0*s1*s0^2, (s0*s1)^2]]
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> a = B([Integer(2), Integer(2), -Integer(1), -Integer(1), Integer(2), Integer(2)]) >>> a.sliding_circuits() [[(s0^-1*s1^-1*s0^-1)^2*s1^3*s0^2*s1^3], [s0^-1*s1^-1*s0^-2*s1^-1*s0^2*s1^2*s0^3], [s0^-1*s1^-1*s0^-2*s1^-1*s0^3*s1^2*s0^2], [(s0^-1*s1^-1*s0^-1)^2*s1^4*s0^2*s1^2], [(s0^-1*s1^-1*s0^-1)^2*s1^2*s0^2*s1^4], [s0^-1*s1^-1*s0^-2*s1^-1*s0*s1^2*s0^4], [(s0^-1*s1^-1*s0^-1)^2*s1^5*s0^2*s1], [s0^-1*s1^-1*s0^-2*s1^-1*s0^4*s1^2*s0], [(s0^-1*s1^-1*s0^-1)^2*s1*s0^2*s1^5], [s0^-1*s1^-1*s0^-2*s1*s0^5], [(s0^-1*s1^-1*s0^-1)^2*s1*s0^6*s1], [s0^-1*s1^-1*s0^-2*s1^5*s0]] >>> b = B([Integer(2), Integer(1), Integer(2), Integer(1)]) >>> b.sliding_circuits() [[s0*s1*s0^2, (s0*s1)^2]]
- sliding_circuits_element()[source]¶
Return an element of the braid’s sliding circuits, and the conjugating braid.
EXAMPLES:
sage: B = BraidGroup(4) sage: b = B([1, 2, 1, 2, 3, -1, 2, -1, 3]) sage: b.sliding_circuits_element() (s0*s1*s0*s2*s1, s0^2*s1*s2)
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b = B([Integer(1), Integer(2), Integer(1), Integer(2), Integer(3), -Integer(1), Integer(2), -Integer(1), Integer(3)]) >>> b.sliding_circuits_element() (s0*s1*s0*s2*s1, s0^2*s1*s2)
- strands()[source]¶
Return the number of strands in the braid.
EXAMPLES:
sage: B = BraidGroup(4) sage: b = B([1, 2, -1, 3, -2]) sage: b.strands() 4
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b = B([Integer(1), Integer(2), -Integer(1), Integer(3), -Integer(2)]) >>> b.strands() 4
- super_summit_set()[source]¶
Return a list with the super summit set of the braid.
EXAMPLES:
sage: B = BraidGroup(3) sage: b = B([1, 2, -1, -2, -2, 1]) sage: b.super_summit_set() [s0^-1*s1^-1*s0^-2*s1^2*s0^2, (s0^-1*s1^-1*s0^-1)^2*s1^2*s0^3*s1, (s0^-1*s1^-1*s0^-1)^2*s1*s0^3*s1^2, s0^-1*s1^-1*s0^-2*s1^-1*s0*s1^3*s0]
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> b = B([Integer(1), Integer(2), -Integer(1), -Integer(2), -Integer(2), Integer(1)]) >>> b.super_summit_set() [s0^-1*s1^-1*s0^-2*s1^2*s0^2, (s0^-1*s1^-1*s0^-1)^2*s1^2*s0^3*s1, (s0^-1*s1^-1*s0^-1)^2*s1*s0^3*s1^2, s0^-1*s1^-1*s0^-2*s1^-1*s0*s1^3*s0]
- super_summit_set_element()[source]¶
Return an element of the braid’s super summit set and the conjugating braid.
EXAMPLES:
sage: B = BraidGroup(4) sage: b = B([1, 2, 1, 2, 3, -1, 2, 1, 3]) sage: b.super_summit_set_element() (s0*s2*s0*s1*s2*s1*s0, s0^-1*s1^-1*s0^-1*s2^-1*s1^-1*s0^-1*s1*s0*s2*s1*s0)
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b = B([Integer(1), Integer(2), Integer(1), Integer(2), Integer(3), -Integer(1), Integer(2), Integer(1), Integer(3)]) >>> b.super_summit_set_element() (s0*s2*s0*s1*s2*s1*s0, s0^-1*s1^-1*s0^-1*s2^-1*s1^-1*s0^-1*s1*s0*s2*s1*s0)
- thurston_type()[source]¶
Return the thurston_type of
self.OUTPUT: one of
'reducible','periodic'or'pseudo-anosov'EXAMPLES:
sage: B = BraidGroup(3) sage: b = B([1, 2, -1]) sage: b.thurston_type() 'reducible' sage: a = B([2, 2, -1, -1, 2, 2]) sage: a.thurston_type() 'pseudo-anosov' sage: c = B([2, 1, 2, 1]) sage: c.thurston_type() 'periodic'
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> b = B([Integer(1), Integer(2), -Integer(1)]) >>> b.thurston_type() 'reducible' >>> a = B([Integer(2), Integer(2), -Integer(1), -Integer(1), Integer(2), Integer(2)]) >>> a.thurston_type() 'pseudo-anosov' >>> c = B([Integer(2), Integer(1), Integer(2), Integer(1)]) >>> c.thurston_type() 'periodic'
- trajectory()[source]¶
Return the braid’s trajectory.
EXAMPLES:
sage: B = BraidGroup(4) sage: b = B([1, 2, 1, 2, 3, -1, 2, -1, 3]) sage: b.trajectory() [s0^-1*s1^-1*s0^-1*s2^-1*s1^-1*s2*s0*s1*s2*s1*s0^2*s1*s2^2, s0*s1*s2^3, s0*s1*s2*s1^2, s0*s1*s0*s2*s1]
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b = B([Integer(1), Integer(2), Integer(1), Integer(2), Integer(3), -Integer(1), Integer(2), -Integer(1), Integer(3)]) >>> b.trajectory() [s0^-1*s1^-1*s0^-1*s2^-1*s1^-1*s2*s0*s1*s2*s1*s0^2*s1*s2^2, s0*s1*s2^3, s0*s1*s2*s1^2, s0*s1*s0*s2*s1]
- tropical_coordinates()[source]¶
Return the tropical coordinates of
selfin the braid group \(B_n\).OUTPUT: list of \(2n\) tropical integers
EXAMPLES:
sage: B = BraidGroup(3) sage: b = B([1]) sage: tc = b.tropical_coordinates(); tc [1, 0, 0, 2, 0, 1] sage: tc[0].parent() Tropical semiring over Integer Ring sage: b = B([-2, -2, -1, -1, 2, 2, 1, 1]) sage: b.tropical_coordinates() [1, -19, -12, 9, 0, 13]
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> b = B([Integer(1)]) >>> tc = b.tropical_coordinates(); tc [1, 0, 0, 2, 0, 1] >>> tc[Integer(0)].parent() Tropical semiring over Integer Ring >>> b = B([-Integer(2), -Integer(2), -Integer(1), -Integer(1), Integer(2), Integer(2), Integer(1), Integer(1)]) >>> b.tropical_coordinates() [1, -19, -12, 9, 0, 13]
REFERENCES:
- ultra_summit_set()[source]¶
Return a list with the orbits of the ultra summit set of
self.EXAMPLES:
sage: B = BraidGroup(3) sage: a = B([2, 2, -1, -1, 2, 2]) sage: b = B([2, 1, 2, 1]) sage: b.ultra_summit_set() [[s0*s1*s0^2, (s0*s1)^2]] sage: a.ultra_summit_set() [[(s0^-1*s1^-1*s0^-1)^2*s1^3*s0^2*s1^3, (s0^-1*s1^-1*s0^-1)^2*s1^2*s0^2*s1^4, (s0^-1*s1^-1*s0^-1)^2*s1*s0^2*s1^5, s0^-1*s1^-1*s0^-2*s1^5*s0, (s0^-1*s1^-1*s0^-1)^2*s1^5*s0^2*s1, (s0^-1*s1^-1*s0^-1)^2*s1^4*s0^2*s1^2], [s0^-1*s1^-1*s0^-2*s1^-1*s0^2*s1^2*s0^3, s0^-1*s1^-1*s0^-2*s1^-1*s0*s1^2*s0^4, s0^-1*s1^-1*s0^-2*s1*s0^5, (s0^-1*s1^-1*s0^-1)^2*s1*s0^6*s1, s0^-1*s1^-1*s0^-2*s1^-1*s0^4*s1^2*s0, s0^-1*s1^-1*s0^-2*s1^-1*s0^3*s1^2*s0^2]]
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> a = B([Integer(2), Integer(2), -Integer(1), -Integer(1), Integer(2), Integer(2)]) >>> b = B([Integer(2), Integer(1), Integer(2), Integer(1)]) >>> b.ultra_summit_set() [[s0*s1*s0^2, (s0*s1)^2]] >>> a.ultra_summit_set() [[(s0^-1*s1^-1*s0^-1)^2*s1^3*s0^2*s1^3, (s0^-1*s1^-1*s0^-1)^2*s1^2*s0^2*s1^4, (s0^-1*s1^-1*s0^-1)^2*s1*s0^2*s1^5, s0^-1*s1^-1*s0^-2*s1^5*s0, (s0^-1*s1^-1*s0^-1)^2*s1^5*s0^2*s1, (s0^-1*s1^-1*s0^-1)^2*s1^4*s0^2*s1^2], [s0^-1*s1^-1*s0^-2*s1^-1*s0^2*s1^2*s0^3, s0^-1*s1^-1*s0^-2*s1^-1*s0*s1^2*s0^4, s0^-1*s1^-1*s0^-2*s1*s0^5, (s0^-1*s1^-1*s0^-1)^2*s1*s0^6*s1, s0^-1*s1^-1*s0^-2*s1^-1*s0^4*s1^2*s0, s0^-1*s1^-1*s0^-2*s1^-1*s0^3*s1^2*s0^2]]
- ultra_summit_set_element()[source]¶
Return an element of the braid’s ultra summit set and the conjugating braid.
EXAMPLES:
sage: B = BraidGroup(4) sage: b = B([1, 2, 1, 2, 3, -1, 2, -1, 3]) sage: b.ultra_summit_set_element() (s0*s1*s0*s2*s1, s0^-1*s1^-1*s0^-1*s2^-1*s1^-1*s0^-1*s1*s2*s1^2*s0)
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> b = B([Integer(1), Integer(2), Integer(1), Integer(2), Integer(3), -Integer(1), Integer(2), -Integer(1), Integer(3)]) >>> b.ultra_summit_set_element() (s0*s1*s0*s2*s1, s0^-1*s1^-1*s0^-1*s2^-1*s1^-1*s0^-1*s1*s2*s1^2*s0)
- sage.groups.braid.BraidGroup(n=None, names='s')[source]¶
Construct a Braid Group.
INPUT:
n– integer orNone(default). The number of strands. If not specified thenamesare counted and the group is assumed to have one more strand than generators.names– string or list/tuple/iterable of strings (default:'x'); the generator names or name prefix
EXAMPLES:
sage: B.<a,b> = BraidGroup(); B Braid group on 3 strands sage: H = BraidGroup('a, b') sage: B is H True sage: BraidGroup(3) Braid group on 3 strands
>>> from sage.all import * >>> B = BraidGroup(names=('a', 'b',)); (a, b,) = B._first_ngens(2); B Braid group on 3 strands >>> H = BraidGroup('a, b') >>> B is H True >>> BraidGroup(Integer(3)) Braid group on 3 strands
The entry can be either a string with the names of the generators, or the number of generators and the prefix of the names to be given. The default prefix is
's'sage: B = BraidGroup(3); B.generators() (s0, s1) sage: BraidGroup(3, 'g').generators() (g0, g1)
>>> from sage.all import * >>> B = BraidGroup(Integer(3)); B.generators() (s0, s1) >>> BraidGroup(Integer(3), 'g').generators() (g0, g1)
Since the word problem for the braid groups is solvable, their Cayley graph can be locally obtained as follows (see Issue #16059):
sage: def ball(group, radius): ....: ret = set() ....: ret.add(group.one()) ....: for length in range(1, radius): ....: for w in Words(alphabet=group.gens(), length=length): ....: ret.add(prod(w)) ....: return ret sage: B = BraidGroup(4) sage: GB = B.cayley_graph(elements=ball(B, 4), generators=B.gens()); GB # needs sage.combinat sage.graphs Digraph on 31 vertices
>>> from sage.all import * >>> def ball(group, radius): ... ret = set() ... ret.add(group.one()) ... for length in range(Integer(1), radius): ... for w in Words(alphabet=group.gens(), length=length): ... ret.add(prod(w)) ... return ret >>> B = BraidGroup(Integer(4)) >>> GB = B.cayley_graph(elements=ball(B, Integer(4)), generators=B.gens()); GB # needs sage.combinat sage.graphs Digraph on 31 vertices
Since the braid group has nontrivial relations, this graph contains less vertices than the one associated to the free group (which is a tree):
sage: F = FreeGroup(3) sage: GF = F.cayley_graph(elements=ball(F, 4), generators=F.gens()); GF # needs sage.combinat sage.graphs Digraph on 40 vertices
>>> from sage.all import * >>> F = FreeGroup(Integer(3)) >>> GF = F.cayley_graph(elements=ball(F, Integer(4)), generators=F.gens()); GF # needs sage.combinat sage.graphs Digraph on 40 vertices
- class sage.groups.braid.BraidGroup_class(names)[source]¶
Bases:
FiniteTypeArtinGroupThe braid group on \(n\) strands.
EXAMPLES:
sage: B1 = BraidGroup(5) sage: B1 Braid group on 5 strands sage: B2 = BraidGroup(3) sage: B1==B2 False sage: B2 is BraidGroup(3) True
>>> from sage.all import * >>> B1 = BraidGroup(Integer(5)) >>> B1 Braid group on 5 strands >>> B2 = BraidGroup(Integer(3)) >>> B1==B2 False >>> B2 is BraidGroup(Integer(3)) True
- TL_basis_with_drain(drain_size)[source]¶
Return a basis of a summand of the Temperley–Lieb–Jones representation of
self.The basis elements are given by non-intersecting pairings of \(n+d\) points in a square with \(n\) points marked ‘on the top’ and \(d\) points ‘on the bottom’ so that every bottom point is paired with a top point. Here, \(n\) is the number of strands of the braid group, and \(d\) is specified by
drain_size.A basis element is specified as a list of integers obtained by considering the pairings as obtained as the ‘highest term’ of trivalent trees marked by Jones–Wenzl projectors (see e.g. [Wan2010]). In practice, this is a list of nonnegative integers whose first element is
drain_size, whose last element is \(0\), and satisfying that consecutive integers have difference \(1\). Moreover, the length of each basis element is \(n + 1\).Given these rules, the list of lists is constructed recursively in the natural way.
INPUT:
drain_size– integer between 0 and the number of strands (both inclusive)
OUTPUT: list of basis elements, each of which is a list of integers
EXAMPLES:
We calculate the basis for the appropriate vector space for \(B_5\) when \(d = 3\):
sage: B = BraidGroup(5) sage: B.TL_basis_with_drain(3) [[3, 4, 3, 2, 1, 0], [3, 2, 3, 2, 1, 0], [3, 2, 1, 2, 1, 0], [3, 2, 1, 0, 1, 0]]
>>> from sage.all import * >>> B = BraidGroup(Integer(5)) >>> B.TL_basis_with_drain(Integer(3)) [[3, 4, 3, 2, 1, 0], [3, 2, 3, 2, 1, 0], [3, 2, 1, 2, 1, 0], [3, 2, 1, 0, 1, 0]]
The number of basis elements hopefully corresponds to the general formula for the dimension of the representation spaces:
sage: B = BraidGroup(10) sage: d = 2 sage: B.dimension_of_TL_space(d) == len(B.TL_basis_with_drain(d)) True
>>> from sage.all import * >>> B = BraidGroup(Integer(10)) >>> d = Integer(2) >>> B.dimension_of_TL_space(d) == len(B.TL_basis_with_drain(d)) True
- TL_representation(drain_size, variab=None)[source]¶
Return representation matrices of the Temperley–Lieb–Jones representation of standard braid group generators and inverses of
self.The basis is given by non-intersecting pairings of \((n+d)\) points, where \(n\) is the number of strands, and \(d\) is given by
drain_size, and the pairings satisfy certain rules. SeeTL_basis_with_drain()for details. This basis has the useful property that all resulting entries can be regarded as Laurent polynomials.We use the convention that the eigenvalues of the standard generators are \(1\) and \(-A^4\), where \(A\) is the generator of the Laurent polynomial ring.
When \(d = n - 2\) and the variables are picked appropriately, the resulting representation is equivalent to the reduced Burau representation. When \(d = n\), the resulting representation is trivial and 1-dimensional.
INPUT:
drain_size– integer between 0 and the number of strands (both inclusive)variab– variable (default:None); the variable in the entries of the matrices; ifNone, then use a default variable in \(\ZZ[A,A^{-1}]\)
OUTPUT: list of matrices corresponding to the representations of each of the standard generators and their inverses
EXAMPLES:
sage: B = BraidGroup(4) sage: B.TL_representation(0) [( [ 1 0] [ 1 0] [ A^2 -A^4], [ A^-2 -A^-4] ), ( [-A^4 A^2] [-A^-4 A^-2] [ 0 1], [ 0 1] ), ( [ 1 0] [ 1 0] [ A^2 -A^4], [ A^-2 -A^-4] )] sage: R.<A> = LaurentPolynomialRing(GF(2)) sage: B.TL_representation(0, variab=A) [( [ 1 0] [ 1 0] [A^2 A^4], [A^-2 A^-4] ), ( [A^4 A^2] [A^-4 A^-2] [ 0 1], [ 0 1] ), ( [ 1 0] [ 1 0] [A^2 A^4], [A^-2 A^-4] )] sage: B = BraidGroup(8) sage: B.TL_representation(8) [([1], [1]), ([1], [1]), ([1], [1]), ([1], [1]), ([1], [1]), ([1], [1]), ([1], [1])]
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> B.TL_representation(Integer(0)) [( [ 1 0] [ 1 0] [ A^2 -A^4], [ A^-2 -A^-4] ), ( [-A^4 A^2] [-A^-4 A^-2] [ 0 1], [ 0 1] ), ( [ 1 0] [ 1 0] [ A^2 -A^4], [ A^-2 -A^-4] )] >>> R = LaurentPolynomialRing(GF(Integer(2)), names=('A',)); (A,) = R._first_ngens(1) >>> B.TL_representation(Integer(0), variab=A) [( [ 1 0] [ 1 0] [A^2 A^4], [A^-2 A^-4] ), ( [A^4 A^2] [A^-4 A^-2] [ 0 1], [ 0 1] ), ( [ 1 0] [ 1 0] [A^2 A^4], [A^-2 A^-4] )] >>> B = BraidGroup(Integer(8)) >>> B.TL_representation(Integer(8)) [([1], [1]), ([1], [1]), ([1], [1]), ([1], [1]), ([1], [1]), ([1], [1]), ([1], [1])]
- an_element()[source]¶
Return an element of the braid group.
This is used both for illustration and testing purposes.
EXAMPLES:
sage: B = BraidGroup(2) sage: B.an_element() s
>>> from sage.all import * >>> B = BraidGroup(Integer(2)) >>> B.an_element() s
- as_permutation_group()[source]¶
Return an isomorphic permutation group.
OUTPUT: this raises a
ValueErrorerror since braid groups are infinite
- dimension_of_TL_space(drain_size)[source]¶
Return the dimension of a particular Temperley–Lieb representation summand of
self.Following the notation of
TL_basis_with_drain(), the summand is the one corresponding to the number of drains being fixed to bedrain_size.INPUT:
drain_size– integer between 0 and the number of strands (both inclusive)
EXAMPLES:
Calculation of the dimension of the representation of \(B_8\) corresponding to having \(2\) drains:
sage: B = BraidGroup(8) sage: B.dimension_of_TL_space(2) 28
>>> from sage.all import * >>> B = BraidGroup(Integer(8)) >>> B.dimension_of_TL_space(Integer(2)) 28
The direct sum of endomorphism spaces of these vector spaces make up the entire Temperley–Lieb algebra:
sage: import sage.combinat.diagram_algebras as da # needs sage.combinat sage: B = BraidGroup(6) sage: dimensions = [B.dimension_of_TL_space(d)**2 for d in [0, 2, 4, 6]] sage: total_dim = sum(dimensions) sage: total_dim == len(list(da.temperley_lieb_diagrams(6))) # long time, needs sage.combinat True
>>> from sage.all import * >>> import sage.combinat.diagram_algebras as da # needs sage.combinat >>> B = BraidGroup(Integer(6)) >>> dimensions = [B.dimension_of_TL_space(d)**Integer(2) for d in [Integer(0), Integer(2), Integer(4), Integer(6)]] >>> total_dim = sum(dimensions) >>> total_dim == len(list(da.temperley_lieb_diagrams(Integer(6)))) # long time, needs sage.combinat True
- epimorphisms(H)[source]¶
Return the epimorphisms from
selftoH, up to automorphism of \(H\) passing through thetwo generator presentationofself.INPUT:
H– another group
EXAMPLES:
sage: B = BraidGroup(5) sage: B.epimorphisms(SymmetricGroup(5)) [Generic morphism: From: Braid group on 5 strands To: Symmetric group of order 5! as a permutation group Defn: s0 |--> (1,5) s1 |--> (4,5) s2 |--> (3,4) s3 |--> (2,3)]
>>> from sage.all import * >>> B = BraidGroup(Integer(5)) >>> B.epimorphisms(SymmetricGroup(Integer(5))) [Generic morphism: From: Braid group on 5 strands To: Symmetric group of order 5! as a permutation group Defn: s0 |--> (1,5) s1 |--> (4,5) s2 |--> (3,4) s3 |--> (2,3)]
ALGORITHM:
Uses libgap’s GQuotients function.
- mapping_class_action(F)[source]¶
Return the action of
selfin the free group F as mapping class group.This action corresponds to the action of the braid over the punctured disk, whose fundamental group is the free group on as many generators as strands.
In Sage, this action is the result of multiplying a free group element with a braid. So you generally do not have to construct this action yourself.
OUTPUT: a
MappingClassGroupActionEXAMPLES:
sage: B = BraidGroup(3) sage: B.inject_variables() Defining s0, s1 sage: F.<a,b,c> = FreeGroup(3) sage: A = B.mapping_class_action(F) sage: A(a,s0) a*b*a^-1 sage: a * s0 # simpler notation a*b*a^-1
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> B.inject_variables() Defining s0, s1 >>> F = FreeGroup(Integer(3), names=('a', 'b', 'c',)); (a, b, c,) = F._first_ngens(3) >>> A = B.mapping_class_action(F) >>> A(a,s0) a*b*a^-1 >>> a * s0 # simpler notation a*b*a^-1
- mirror_involution()[source]¶
Return the mirror involution of
self.This automorphism maps a braid to another one by replacing each generator in its word by the inverse. In general this is different from the inverse of the braid since the order of the generators in the word is not reversed.
EXAMPLES:
sage: B = BraidGroup(4) sage: mirr = B.mirror_involution() sage: b = B((1,-2,-1,3,2,1)) sage: bm = mirr(b); bm s0^-1*s1*s0*s2^-1*s1^-1*s0^-1 sage: bm == ~b False sage: bm.is_conjugated(b) False sage: bm.is_conjugated(~b) True
>>> from sage.all import * >>> B = BraidGroup(Integer(4)) >>> mirr = B.mirror_involution() >>> b = B((Integer(1),-Integer(2),-Integer(1),Integer(3),Integer(2),Integer(1))) >>> bm = mirr(b); bm s0^-1*s1*s0*s2^-1*s1^-1*s0^-1 >>> bm == ~b False >>> bm.is_conjugated(b) False >>> bm.is_conjugated(~b) True
- presentation_two_generators(isomorphisms=False)[source]¶
Construct a finitely presented group isomorphic to
selfwith only two generators.INPUT:
isomorphism– boolean (default:False); ifTrue, then an isomorphism fromselfand the isomorphic group and its inverse is also returned
EXAMPLES:
sage: B = BraidGroup(3) sage: B.presentation_two_generators() Finitely presented group < x0, x1 | x1^3*x0^-2 > sage: B = BraidGroup(4) sage: G, hom1, hom2 = B.presentation_two_generators(isomorphisms=True) sage: G Finitely presented group < x0, x1 | x1^4*x0^-3, x0*x1*x0*x1^-2*x0^-1*x1^3*x0^-1*x1^-2 > sage: hom1(B.gen(0)) x0*x1^-1 sage: hom1(B.gen(1)) x1*x0*x1^-2 sage: hom1(B.gen(2)) x1^2*x0*x1^-3 sage: all(hom2(hom1(a)) == a for a in B.gens()) True sage: all(hom2(a) == B.one() for a in G.relations()) True
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> B.presentation_two_generators() Finitely presented group < x0, x1 | x1^3*x0^-2 > >>> B = BraidGroup(Integer(4)) >>> G, hom1, hom2 = B.presentation_two_generators(isomorphisms=True) >>> G Finitely presented group < x0, x1 | x1^4*x0^-3, x0*x1*x0*x1^-2*x0^-1*x1^3*x0^-1*x1^-2 > >>> hom1(B.gen(Integer(0))) x0*x1^-1 >>> hom1(B.gen(Integer(1))) x1*x0*x1^-2 >>> hom1(B.gen(Integer(2))) x1^2*x0*x1^-3 >>> all(hom2(hom1(a)) == a for a in B.gens()) True >>> all(hom2(a) == B.one() for a in G.relations()) True
- some_elements()[source]¶
Return a list of some elements of the braid group.
This is used both for illustration and testing purposes.
EXAMPLES:
sage: B = BraidGroup(3) sage: B.some_elements() [s0, s0*s1, (s0*s1)^3]
>>> from sage.all import * >>> B = BraidGroup(Integer(3)) >>> B.some_elements() [s0, s0*s1, (s0*s1)^3]
- class sage.groups.braid.MappingClassGroupAction(G, M)[source]¶
Bases:
ActionThe right action of the braid group the free group as the mapping class group of the punctured disk.
That is, this action is the action of the braid over the punctured disk, whose fundamental group is the free group on as many generators as strands.
This action is defined as follows:
\[\begin{split}x_j \cdot \sigma_i=\begin{cases} x_{j}\cdot x_{j+1}\cdot {x_j}^{-1} & \text{if $i=j$} \\ x_{j-1} & \text{if $i=j-1$} \\ x_{j} & \text{otherwise} \end{cases},\end{split}\]where \(\sigma_i\) are the generators of the braid group on \(n\) strands, and \(x_j\) the generators of the free group of rank \(n\).
You should left multiplication of the free group element by the braid to compute the action. Alternatively, use the
mapping_class_action()method of the braid group to construct this action.EXAMPLES:
sage: B.<s0,s1,s2> = BraidGroup(4) sage: F.<x0,x1,x2,x3> = FreeGroup(4) sage: x0 * s1 x0 sage: x1 * s1 x1*x2*x1^-1 sage: x1^-1 * s1 x1*x2^-1*x1^-1 sage: A = B.mapping_class_action(F) sage: A Right action by Braid group on 4 strands on Free Group on generators {x0, x1, x2, x3} sage: A(x0, s1) x0 sage: A(x1, s1) x1*x2*x1^-1 sage: A(x1^-1, s1) x1*x2^-1*x1^-1
>>> from sage.all import * >>> B = BraidGroup(Integer(4), names=('s0', 's1', 's2',)); (s0, s1, s2,) = B._first_ngens(3) >>> F = FreeGroup(Integer(4), names=('x0', 'x1', 'x2', 'x3',)); (x0, x1, x2, x3,) = F._first_ngens(4) >>> x0 * s1 x0 >>> x1 * s1 x1*x2*x1^-1 >>> x1**-Integer(1) * s1 x1*x2^-1*x1^-1 >>> A = B.mapping_class_action(F) >>> A Right action by Braid group on 4 strands on Free Group on generators {x0, x1, x2, x3} >>> A(x0, s1) x0 >>> A(x1, s1) x1*x2*x1^-1 >>> A(x1**-Integer(1), s1) x1*x2^-1*x1^-1
- class sage.groups.braid.RightQuantumWord(words)[source]¶
Bases:
objectA right quantum word as in Definition 4.1 of [HL2018].
INPUT:
words– an element in a suitable free algebra over a Laurent polynomial ring in one variable; this input does not need to be in reduced form, but the monomials for the input can come in any order
EXAMPLES:
sage: from sage.groups.braid import RightQuantumWord sage: fig_8 = BraidGroup(3)([-1, 2, -1, 2]) sage: ( ....: bp_1, cp_1, ap_1, ....: bp_3, cp_3, ap_3, ....: bm_0, cm_0, am_0, ....: bm_2, cm_2, am_2 ....: ) = fig_8.deformed_burau_matrix().parent().base_ring().gens() sage: q = bp_1.base_ring().gen() sage: RightQuantumWord(ap_1*cp_1 + q**3*bm_2*bp_1*am_0*cm_0) The right quantum word represented by q*cp_1*ap_1 + q^2*bp_1*cm_0*am_0*bm_2 reduced from ap_1*cp_1 + q^3*bm_2*bp_1*am_0*cm_0
>>> from sage.all import * >>> from sage.groups.braid import RightQuantumWord >>> fig_8 = BraidGroup(Integer(3))([-Integer(1), Integer(2), -Integer(1), Integer(2)]) >>> ( ... bp_1, cp_1, ap_1, ... bp_3, cp_3, ap_3, ... bm_0, cm_0, am_0, ... bm_2, cm_2, am_2 ... ) = fig_8.deformed_burau_matrix().parent().base_ring().gens() >>> q = bp_1.base_ring().gen() >>> RightQuantumWord(ap_1*cp_1 + q**Integer(3)*bm_2*bp_1*am_0*cm_0) The right quantum word represented by q*cp_1*ap_1 + q^2*bp_1*cm_0*am_0*bm_2 reduced from ap_1*cp_1 + q^3*bm_2*bp_1*am_0*cm_0
- eps(N)[source]¶
Evaluate the map \(\mathcal{E}_N\) for a braid.
INPUT:
N– integer; the number of colors
EXAMPLES:
sage: from sage.groups.braid import RightQuantumWord sage: B = BraidGroup(3) sage: b = B([1,-2,1,2]) sage: db = b.deformed_burau_matrix() sage: q = db.parent().base_ring().base_ring().gen() sage: (bp_0, cp_0, ap_0, ....: bp_2, cp_2, ap_2, ....: bp_3, cp_3, ap_3, ....: bm_1, cm_1, am_1) = db.parent().base_ring().gens() sage: rqw = RightQuantumWord( ....: q^3*bp_2*bp_0*ap_0 + q*ap_3*bm_1*am_1*bp_0) sage: rqw.eps(3) -q^-1 + 2*q - q^5 sage: rqw.eps(2) -1 + 2*q - q^2 + q^3 - q^4
>>> from sage.all import * >>> from sage.groups.braid import RightQuantumWord >>> B = BraidGroup(Integer(3)) >>> b = B([Integer(1),-Integer(2),Integer(1),Integer(2)]) >>> db = b.deformed_burau_matrix() >>> q = db.parent().base_ring().base_ring().gen() >>> (bp_0, cp_0, ap_0, ... bp_2, cp_2, ap_2, ... bp_3, cp_3, ap_3, ... bm_1, cm_1, am_1) = db.parent().base_ring().gens() >>> rqw = RightQuantumWord( ... q**Integer(3)*bp_2*bp_0*ap_0 + q*ap_3*bm_1*am_1*bp_0) >>> rqw.eps(Integer(3)) -q^-1 + 2*q - q^5 >>> rqw.eps(Integer(2)) -1 + 2*q - q^2 + q^3 - q^4
Todo
Parallelize this function, calculating all summands in the sum in parallel.
- reduced_word()[source]¶
Return the (reduced) right quantum word.
OUTPUT: an element in the free algebra
EXAMPLES:
sage: from sage.groups.braid import RightQuantumWord sage: fig_8 = BraidGroup(3)([-1, 2, -1, 2]) sage: ( ....: bp_1, cp_1, ap_1, ....: bp_3, cp_3, ap_3, ....: bm_0, cm_0, am_0, ....: bm_2, cm_2, am_2 ....: ) = fig_8.deformed_burau_matrix().parent().base_ring().gens() sage: q = bp_1.base_ring().gen() sage: qw = RightQuantumWord(ap_1*cp_1 + ....: q**3*bm_2*bp_1*am_0*cm_0) sage: qw.reduced_word() q*cp_1*ap_1 + q^2*bp_1*cm_0*am_0*bm_2
>>> from sage.all import * >>> from sage.groups.braid import RightQuantumWord >>> fig_8 = BraidGroup(Integer(3))([-Integer(1), Integer(2), -Integer(1), Integer(2)]) >>> ( ... bp_1, cp_1, ap_1, ... bp_3, cp_3, ap_3, ... bm_0, cm_0, am_0, ... bm_2, cm_2, am_2 ... ) = fig_8.deformed_burau_matrix().parent().base_ring().gens() >>> q = bp_1.base_ring().gen() >>> qw = RightQuantumWord(ap_1*cp_1 + ... q**Integer(3)*bm_2*bp_1*am_0*cm_0) >>> qw.reduced_word() q*cp_1*ap_1 + q^2*bp_1*cm_0*am_0*bm_2
Todo
Parallelize this function, calculating all summands in the sum in parallel.
- tuples()[source]¶
Get a representation of the right quantum word as a
dict, with keys monomials in the free algebra represented as tuples and values in elements the Laurent polynomial ring in one variable.This is in the reduced form as outlined in Definition 4.1 of [HL2018].
OUTPUT: a dict of tuples of ints corresponding to the exponents in the generators with values in the algebra’s base ring
EXAMPLES:
sage: from sage.groups.braid import RightQuantumWord sage: fig_8 = BraidGroup(3)([-1, 2, -1, 2]) sage: ( ....: bp_1, cp_1, ap_1, ....: bp_3, cp_3, ap_3, ....: bm_0, cm_0, am_0, ....: bm_2, cm_2, am_2 ....: ) = fig_8.deformed_burau_matrix().parent().base_ring().gens() sage: q = bp_1.base_ring().gen() sage: qw = RightQuantumWord(ap_1*cp_1 + ....: q**3*bm_2*bp_1*am_0*cm_0) sage: for key, value in qw.tuples.items(): ....: print(key, value) ....: (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0) q (1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0) q^2
>>> from sage.all import * >>> from sage.groups.braid import RightQuantumWord >>> fig_8 = BraidGroup(Integer(3))([-Integer(1), Integer(2), -Integer(1), Integer(2)]) >>> ( ... bp_1, cp_1, ap_1, ... bp_3, cp_3, ap_3, ... bm_0, cm_0, am_0, ... bm_2, cm_2, am_2 ... ) = fig_8.deformed_burau_matrix().parent().base_ring().gens() >>> q = bp_1.base_ring().gen() >>> qw = RightQuantumWord(ap_1*cp_1 + ... q**Integer(3)*bm_2*bp_1*am_0*cm_0) >>> for key, value in qw.tuples.items(): ... print(key, value) ....: (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0) q (1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0) q^2