In this notebook we verify that all links (except the 5 stated in Proposition 3.1) in the two parameter family K_nm are hyperbolic. We defined K_nm via the surgery description L12n1739(-1,2n+m+3)(0,0)(1,n+1). (The different signs stem from the fact that L12n1739 is the mirror of the surgery description from our paper.) Now the idea is to use Gromov-Thurston's 2pi-Theorem to see that all long fillings are directly hyperbolic and check the remaining short fillings directly.

import snappy

L=snappy.Manifold('L12n1739')
L.verify_hyperbolicity()[0]

L.dehn_fill([(-1,5),(0,0),(1,2)])
L.identify()

L.is_isometric_to(snappy.Manifold('o9_30634'),return_isometries=True)

L=snappy.Manifold('L12n1739')
L.short_slopes(length=6.3,verified=True)

Thus it follows that if (n+1 is not -2,-1,0,1,2) and (2n+m+3 is not -3,-2,-1,0,1,2) then K_nm is hyperbolic.

In the other cases, we first observe that L(0,0(0,0)(1,0) is the T(4,2) torus link. And thus L(1,k)(0,0)(1,0) is the T(-2k-1,2) torus knot.

For the remaining short slopes we first fill one remaining component permanently and then use the 2pi-Theorem again to get a finite list of slopes that might not be hyperbolic.

#The short reciprocal integer slopes of cusps 0 and 2.
slopes0=[(-1, 1), (1, 1), (-1, 2), (1, 2), (-1, 3),(1,0)]
slopes2=[(-1, 1), (1, 1), (-1, 2), (1, 2)]

unclear_slopes=[]

for s0 in slopes0:
    L.dehn_fill(s0,0)
    F=L.filled_triangulation()
    short=F.short_slopes(length=6.3,verified=True)[1]
    for s2 in short:
        if abs(s2[0])==1:
            if abs(s2[1])!=0:
                unclear_slopes.append([s0,(0,0),s2])
L.dehn_fill((0,0),0)

for s2 in slopes2:
    L.dehn_fill(s2,2)
    F=L.filled_triangulation()
    short=F.short_slopes(length=6.3,verified=True)[0]
    for s0 in short:
        if abs(s0[0])==1:
            if abs(s0[1])!=0:
                unclear_slopes.append([s0,(0,0),s2])
L.dehn_fill((0,0),2)

unclear_slopes

len(unclear_slopes)

We are left with checking these 46 slopes. We first try to verify that they are hyperbolic via SnapPy.

def all_positive(manifold):
    '''
    Checks if the solution type of a triangulation is positive.
    '''
    return manifold.solution_type() == 'all tetrahedra positively oriented'

def find_positive_triangulations(manifold,number=1,tries=100):
    '''
    Searches for one triangulation with a positive solution type.
    (Or if number is set to a different value also for different such triangulations.)
    '''
    M = manifold.copy()
    pos_triangulations=[]
    for i in range(tries):
        if all_positive(M):
            pos_triangulations.append(M)
            if len(pos_triangulations)==number:
                return pos_triangulations
            break
        M.randomize()
    for d in M.dual_curves(max_segments=500):
        X = M.drill(d)
        X = X.filled_triangulation()
        X.dehn_fill((1,0),-1)
        for i in range(tries):
            if all_positive(X):
                pos_triangulations.append(X)
                if len(pos_triangulations)==number:
                    return pos_triangulations
                break
            X.randomize()

    # In the closed case, here is another trick.
    if all(not c for c in M.cusp_info('is_complete')):
        for i in range(tries):
            # Drills out a random edge
            X = M.__class__(M.filled_triangulation())
            if all_positive(X):
                pos_triangulations.append(X)
                if len(pos_triangulations)==number:
                    return pos_triangulations
            break
            M.randomize()
    return pos_triangulations

still_unclear=[]
for s in unclear_slopes:
    L.dehn_fill(s)
    try:
        P=find_positive_triangulations(L,number=1,tries=100)[0]
    except (IndexError,snappy.SnapPeaFatalError):
        P=L
    hyp=P.verify_hyperbolicity()[0]
    print(L,hyp)
    if hyp==False:
        still_unclear.append(s)

still_unclear

Thus we are left with 13 knots in the family which are probably not hyperbolic.

Actually, SnapPy recognizes all of them as hyperbolic or torus knots.

for s in still_unclear:
    L.dehn_fill(s)
    F=L.filled_triangulation()
    print(L,F.identify())

All the above knots are torus knots (or hyperbolic in the case of K12n242):

K5a2 = T(5,2)

K8n3 = T(4,3)

K10n21 = T(5,3)

In total we have shown the statement from Proposition 3.1.(1): K_nm is hyperbolic for any (n,m) except for (n,m)=(-1,k) for k an integer, or (n,m)=(0,0),(0,-1),(0,-2),(-2,1),(-2,0),(-2,-1)

And in all the above non-hyperbolic cases K_nm is a torus knot.