In this notebook we will prove that o9_30634 is not braid positive.

First, we verify the claimed braid word.

In [1]:
import snappy

word=[2,1,3,2,2,1,3,2,2,1,3,2,-1,2,1,1,2]
B=snappy.Link(braid_closure=word).exterior()
B.identify()
Out[1]:
[o9_30634(0,0), K9_449(0,0)]

We make sure that the isomorphism from B to K9_449 is really orientation preserving.

Remark: Here we are working with the old version of SnapPy. The new version from April 2021 has a different convention for the braid generators and will yield the mirror of K9_449 (which can then be adjusted to take the inverse of any braid generator).

In [2]:
K=snappy.Manifold('K9_449')
B.is_isometric_to(K,return_isometries=True)
Out[2]:
[0 -> 0
 [1 0] 
 [0 1] 
 Extends to link,
 0 -> 0 
 [-1  0]
 [ 0 -1]
 Extends to link]

We show that B is not braid positive by computing Ito's normalized version of the HOMFLY polynomial and checking that it is not positive.

In [3]:
var('a,b,v,z')

R = PolynomialRing(ZZ, ['a','b'])

def HOMFLY(word):
    '''
    Computes the HOMFLY polynomial.
    '''
    K=snappy.Link(braid_closure=word)
    K.simplify(mode='global')
    L=K.sage_link()
    h(v,z) = L.homfly_polynomial('v', 'z', 'az')
    return h(v,z)

def normalized_HOMFLY(word):
    '''
    Given the braid word of an L-space knot. Returns Ito's normalized version of the HOMFLY.
    '''
    L=snappy.Link(braid_closure=word)
    alex=L.exterior().alexander_polynomial()
    g=alex.degree()/2
    h(v,z)=HOMFLY(word)
    homfly=(h((-a)^(-1/2),b^(1/2))*(-a)^(-g)).expand()
    return homfly

Ito's normalized version of the HOMFLY polynomial is:

In [4]:
normalized_HOMFLY(word)
Out[4]:
a*b^5 + b^6 + a^2*b^3 + 11*a*b^4 + 12*b^5 - a^3*b + 6*a^2*b^2 + 45*a*b^3 + 55*b^4 - a^3 + 10*a^2*b + 83*a*b^2 + 121*b^3 + 4*a^2 + 66*a*b + 133*b^2 + 17*a + 69*b + 13
In [5]:
# A matrix of the coeffcients:
print(matrix(R(normalized_HOMFLY(word)).dict()))

[ 13  69 133 121  55  12   1]
[ 17  66  83  45  11   1   0]
[  4  10   6   1   0   0   0]
[ -1  -1   0   0   0   0   0]

We see that it has two negative coefficients and thus the knot cannot be braid positive.