Here we will show that not all topological volumes of lens spaces are realized by P and W. For that we will get upper bounds on the volume of lens spaces by checking Dunfield's list and then in a second step check which of those manifolds can be realized by surgery on W or P.

In [1]:
import snappy
import csv
In [2]:
exceptional_fillings=[]
with open('exceptional_fillings.csv', 'r') as file:
    reader = csv.reader(file)
    for row in reader:
        exceptional_fillings.append(row)
len(exceptional_fillings)
Out[2]:
205823
In [3]:
lens_spaces=[]
for x in exceptional_fillings:
    if x[1]=='lens_space':
        lens_spaces.append(x)
len(lens_spaces)
Out[3]:
44487
In [4]:
knots_with_lens_space_surgeries=[]
for x in lens_spaces:
    if x[-2] not in knots_with_lens_space_surgeries:
        knots_with_lens_space_surgeries.append(x[-2])
len(knots_with_lens_space_surgeries)
Out[4]:
38056

This is a list of census knots with lens space surgeries. We describe the parent diagram of W and P.

In [5]:
def FKP_bound(vol_surgery_link,lower_volume_bound):
    '''
    Returns the Futer-Kalfagianni-Purcell bound, that gives a lower bound on the normalized 
    length of a slope to ensure that a filling with such a slope on a link with volume = vol_surgery_link 
    has volume at least lower_volume_bound.
    '''
    return 2*3.14159/sqrt(1-(lower_volume_bound/vol_surgery_link)^(2/3))+0.1

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

def better_volume(M,index=100,try_hard=False):
    '''Computes the verified volume. Returns 0 if SnapPy could not do it.'''
    count=0
    while count<index:
        try:
            return M.volume(verified=True)
        except:
            M.randomize()
            count=count+1
    if try_hard==True:
        pos_triang=find_positive_triangulations(M,number=index,tries=index)
        for X in pos_triang:
            vol=better_volume(X,index)
            if vol!=0:
                return vol
    return 0
In [6]:
P=snappy.Manifold('m125')
W=snappy.Manifold('m129')
volP=better_volume(P)
volW=better_volume(W)
print(volW,volP)
3.663862376709? 3.663862376709?
In [7]:
might_be_minimizer=[]

for name in knots_with_lens_space_surgeries:
    K=snappy.Manifold(name)
    vol=better_volume(K)
    if vol<volP:
        might_be_minimizer.append([name,vol])
In [8]:
might_be_minimizer.sort(key=lambda x: x[1])
len(might_be_minimizer)
Out[8]:
1276
In [9]:
might_be_minimizer
Out[9]:
[['m003', 2.0298832128193?],
 ['m006', 2.5689706009367?],
 ['m007', 2.5689706009367?],
 ['m009', 2.666744783449?],
 ['m010', 2.666744783449?],
 ['m011', 2.7818339123961?],
 ['m016', 2.8281220883308?],
 ['m017', 2.8281220883308?],
 ['m019', 2.9441064866767?],
 ['m022', 2.989120282930?],
 ['m023', 2.989120282930?],
 ['m026', 3.0593380577790?],
 ['m027', 3.121334773013?],
 ['m029', 3.148509826441?],
 ['m030', 3.148509826441?],
 ['m033', 3.163963228884?],
 ['m034', 3.166333321250?],
 ['m035', 3.177293278601?],
 ['m036', 3.177293278601?],
 ['m037', 3.177293278601?],
 ['m038', 3.177293278601?],
 ['m039', 3.177293278601?],
 ['m040', 3.177293278601?],
 ['m043', 3.252908048472?],
 ['m044', 3.275676560025?],
 ['m045', 3.275871643944?],
 ['m046', 3.275871643944?],
 ['m047', 3.277062185134?],
 ['s000', 3.296902414327?],
 ['s001', 3.296902414327?],
 ['m049', 3.300217628536?],
 ['m052', 3.3082415547304?],
 ['m054', 3.3317442316411?],
 ['m055', 3.3371917200070?],
 ['m058', 3.356692845142?],
 ['m059', 3.35669284515?],
 ['m060', 3.362093204427?],
 ['m061', 3.366729420471?],
 ['m062', 3.366729420470?],
 ['v0000', 3.37759740823?],
 ['v0001', 3.37759740823?],
 ['m064', 3.3805053992016?],
 ['m066', 3.394540517062?],
 ['m067', 3.394540517062?],
 ['s002', 3.394556686696?],
 ['m069', 3.402991251167?],
 ['m070', 3.402991251167?],
 ['s003', 3.404658962502?],
 ['m071', 3.417914837238?],
 ['m072', 3.424503508771?],
 ['m073', 3.427205246274?],
 ['t00000', 3.43454088591?],
 ['t00001', 3.43454088591?],
 ['m076', 3.439592889349?],
 ['m077', 3.439592889349?],
 ['s004', 3.460128933251?],
 ['s005', 3.460128933250?],
 ['m078', 3.460675847482?],
 ['m079', 3.463688556153?],
 ['m080', 3.463688556153?],
 ['m081', 3.464408817290?],
 ['v0002', 3.465465224753?],
 ['v0003', 3.470395731439?],
 ['s006', 3.471287882780?],
 ['s007', 3.471287882780?],
 ['m082', 3.474247761313?],
 ['m083', 3.4744027755531?],
 ['m084', 3.4761739892390?],
 ['m085', 3.476173989239?],
 ['o9_00000', 3.4761739893?],
 ['o9_00001', 3.47617398924?],
 ['m087', 3.481970896073?],
 ['m089', 3.483898578333?],
 ['m090', 3.483898578333?],
 ['m093', 3.486660146295?],
 ['m095', 3.489870168028?],
 ['s008', 3.493881599711?],
 ['s009', 3.49388159971?],
 ['m096', 3.497133287782?],
 ['s010', 3.499922930053?],
 ['s011', 3.502349676519?],
 ['m098', 3.508917187077?],
 ['t00002', 3.511607436422?],
 ['m099', 3.514082912504?],
 ['t00003', 3.51422960392?],
 ['m100', 3.514252058377?],
 ['s012', 3.51776072855?],
 ['s013', 3.51776072855?],
 ['s014', 3.517760728543?],
 ['s015', 3.51776072855?],
 ['v0004', 3.520121007818?],
 ['v0005', 3.520121007818?],
 ['s017', 3.52619599074?],
 ['m102', 3.526448831456?],
 ['m103', 3.528509496738?],
 ['v0006', 3.52916280051?],
 ['v0007', 3.52916280051?],
 ['m104', 3.530259647492?],
 ['m105', 3.530259647492?],
 ['m106', 3.530953642501?],
 ['m108', 3.531328329872?],
 ['s018', 3.537608784552?],
 ['s019', 3.537608784552?],
 ['m110', 3.542395018379?],
 ['m111', 3.542395018379?],
 ['o9_00002', 3.54333030680?],
 ['o9_00003', 3.544822993160?],
 ['v0008', 3.546723526491?],
 ['s020', 3.546843097064?],
 ['v0009', 3.54707484939?],
 ['v0010', 3.54707484939?],
 ['v0011', 3.547777194650?],
 ['s021', 3.549364348400?],
 ['m112', 3.551403873987?],
 ['s022', 3.553819918512?],
 ['m113', 3.553879655101?],
 ['s024', 3.556390252569?],
 ['s025', 3.556390252569?],
 ['t00004', 3.55752488404?],
 ['t00005', 3.55752488404?],
 ['s026', 3.560748498354?],
 ['v0012', 3.561144716085?],
 ['v0013', 3.561144716085?],
 ['s027', 3.563333662903?],
 ['t00006', 3.56445469922?],
 ['t00007', 3.564454699221?],
 ['s028', 3.565845140356?],
 ['s029', 3.565845140355?],
 ['v0014', 3.56746321945?],
 ['v0015', 3.567463219442?],
 ['s030', 3.568087952770?],
 ['s031', 3.568087952770?],
 ['s032', 3.569593344659?],
 ['s033', 3.571078636994?],
 ['s034', 3.571078636994?],
 ['s035', 3.571564489421?],
 ['s036', 3.572505745545?],
 ['v0017', 3.573882540559?],
 ['s037', 3.574367672977?],
 ['s038', 3.574684891148?],
 ['s039', 3.574684891148?],
 ['t00008', 3.575972812121?],
 ['t00009', 3.576487958604?],
 ['v0018', 3.57875868273?],
 ['v0019', 3.57875868273?],
 ['t00010', 3.57875868273?],
 ['t00011', 3.57875868273?],
 ['v0020', 3.578792225830?],
 ['v0021', 3.578792225830?],
 ['m115', 3.581707325569?],
 ['o9_00004', 3.58224237094?],
 ['o9_00005', 3.58224237094?],
 ['s040', 3.582448512878?],
 ['s041', 3.582448512878?],
 ['v0022', 3.584599815125?],
 ['v0023', 3.585606109952?],
 ['s042', 3.587386785776?],
 ['s043', 3.587430067964?],
 ['o9_00006', 3.58752704920?],
 ['o9_00007', 3.58752704920?],
 ['t00012', 3.587712960811?],
 ['t00013', 3.587712960811?],
 ['s044', 3.588089020504?],
 ['s045', 3.58867524377?],
 ['v0024', 3.588913917792?],
 ['s046', 3.589715239710?],
 ['s047', 3.589715239711?],
 ['m116', 3.589901460754?],
 ['s048', 3.590089964041?],
 ['s049', 3.590089964041?],
 ['s050', 3.591213463572?],
 ['s051', 3.592325182488?],
 ['s052', 3.592325182487?],
 ['s053', 3.592485968318?],
 ['s054', 3.593764197088?],
 ['s055', 3.593764197088?],
 ['v0026', 3.59541637583?],
 ['v0027', 3.595416375823?],
 ['o9_00008', 3.59547865355?],
 ['v0028', 3.595660424387?],
 ['o9_00009', 3.59575406241?],
 ['t00014', 3.596317964325?],
 ['t00015', 3.59631796433?],
 ['v0029', 3.59679609928?],
 ['s056', 3.596882244932?],
 ['s057', 3.596882244932?],
 ['v0030', 3.596882244932?],
 ['v0031', 3.596882244932?],
 ['s058', 3.597126148616?],
 ['s059', 3.597843296291?],
 ['o9_00010', 3.59911989094?],
 ['o9_00011', 3.59911989094?],
 ['t00016', 3.600467262785?],
 ['v0032', 3.600472205207?],
 ['s060', 3.601304697412?],
 ['s061', 3.601304697412?],
 ['v0033', 3.601379470629?],
 ['t00018', 3.603194318883?],
 ['t00019', 3.60319431889?],
 ['m117', 3.603885043390?],
 ['v0034', 3.60395268264?],
 ['v0035', 3.603952682638?],
 ['s062', 3.605029182511?],
 ['o9_00012', 3.6051589910?],
 ['o9_00013', 3.60515899092?],
 ['s063', 3.605387507250?],
 ['s064', 3.605535899658?],
 ['s065', 3.605535899657?],
 ['s066', 3.605882111800?],
 ['v0036', 3.60598644149?],
 ['v0037', 3.60598644149?],
 ['s067', 3.606041999166?],
 ['t00020', 3.60669597277?],
 ['t00021', 3.60715510409?],
 ['v0038', 3.60742916921?],
 ['v0039', 3.607429169201?],
 ['t00022', 3.60789736307?],
 ['t00023', 3.60789736307?],
 ['v0040', 3.608193744961?],
 ['t00024', 3.608257953303?],
 ['t00025', 3.608257953303?],
 ['m118', 3.608689061771?],
 ['m119', 3.608689061771?],
 ['v0041', 3.60879230381?],
 ['t00026', 3.609539174545?],
 ['s068', 3.610206881488?],
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 ['v0056', 3.618973982147?],
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 ['t00040', 3.625160673974?],
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 ['v0079', 3.625796417487?],
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 ['t00043', 3.625960933716?],
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 ['v0081', 3.626866720577?],
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 ['o9_00029', 3.62735368675?],
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 ['v0128', 3.638407633913?],
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 ['v0130', 3.639386687137?],
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 ['o9_00295', 3.655912726471?],
 ['o9_00296', 3.65595851544?],
 ['o9_00297', 3.65597266105?],
 ['t00304', 3.656028018950?],
 ['t00305', 3.65603987445?],
 ['o9_00298', 3.656064690958?],
 ['o9_00299', 3.656064690958?],
 ['o9_00300', 3.65610166725?],
 ['o9_00301', 3.656101667242?],
 ['o9_00302', 3.656122714425?],
 ['o9_00303', 3.656122714425?],
 ['o9_00304', 3.656150421468?],
 ['o9_00305', 3.656150421468?],
 ['t00306', 3.656157391015?],
 ['t00307', 3.65616619376?],
 ['t00308', 3.656174137021?],
 ['o9_00306', 3.656174283933?],
 ['t00309', 3.65618685229?],
 ['o9_00307', 3.656189024694?],
 ['o9_00308', 3.656201376903?],
 ['o9_00309', 3.656210501224?],
 ['o9_00310', 3.65621050123?],
 ['o9_00311', 3.65621282906?],
 ['o9_00312', 3.656218172825?],
 ['o9_00313', 3.656218172825?],
 ['o9_00314', 3.65625088696?],
 ['o9_00315', 3.65625088696?],
 ['o9_00316', 3.65626568249?],
 ['o9_00317', 3.65626568249?],
 ['t00310', 3.65628136117?],
 ['t00311', 3.656281361170?],
 ['o9_00318', 3.656302710673?],
 ['o9_00319', 3.65630271068?],
 ['o9_00320', 3.656334264493?],
 ['o9_00321', 3.656334264493?],
 ['o9_00322', 3.656340676021?],
 ['o9_00323', 3.656346070721?],
 ['t00312', 3.656389702648?],
 ['t00313', 3.65640098954?],
 ['t00314', 3.656406969517?],
 ['t00315', 3.65642075299?],
 ['o9_00324', 3.65645284968?],
 ['o9_00325', 3.656452849674?],
 ['o9_00326', 3.65648437906?],
 ['o9_00327', 3.6564843791?],
 ['o9_00328', 3.65654277282?],
 ['o9_00329', 3.656542772815?],
 ['o9_00330', 3.656561155170?],
 ['o9_00331', 3.65656115517?],
 ['o9_00332', 3.656604253664?],
 ['o9_00333', 3.656614144036?],
 ['o9_00334', 3.656695987037?],
 ['o9_00335', 3.656695987037?],
 ['o9_00336', 3.65670150663?],
 ['o9_00337', 3.656701506629?],
 ['o9_00338', 3.656709105191?],
 ['o9_00339', 3.656721823815?],
 ['o9_00340', 3.656776420232?],
 ['o9_00341', 3.656788810297?],
 ['o9_00342', 3.65687333953?],
 ['o9_00343', 3.65687333953?],
 ['o9_00344', 3.65687778078?],
 ['o9_00345', 3.65687778078?],
 ['o9_00346', 3.656892371459?],
 ['o9_00347', 3.65690083996?],
 ['o9_00348', 3.656900839954?],
 ['o9_00349', 3.656903603933?],
 ['o9_00350', 3.65695101466?],
 ['o9_00351', 3.656957006416?],
 ['o9_00352', 3.656971850342?],
 ['o9_00353', 3.65698075996?],
 ['o9_00354', 3.65701866791?],
 ['o9_00355', 3.65701866791?],
 ['o9_00356', 3.65702704186?],
 ['o9_00357', 3.65703432303?],
 ['o9_00358', 3.65704571110?],
 ['o9_00359', 3.65704571110?],
 ['o9_00360', 3.657049756477?],
 ['o9_00361', 3.657053522613?],
 ['o9_00362', 3.65705352262?],
 ['o9_00363', 3.657059826130?],
 ['t00316', 3.65709889290?],
 ['t00317', 3.65709889290?],
 ['o9_00364', 3.657120210102?],
 ['o9_00365', 3.65712478548?],
 ['o9_00366', 3.65712478549?],
 ['o9_00367', 3.657131359228?],
 ['o9_00368', 3.657139806285?],
 ['o9_00369', 3.65714961691?],
 ['o9_00370', 3.65722826139?],
 ['o9_00371', 3.65722826140?],
 ['o9_00372', 3.657234197906?],
 ['o9_00373', 3.65724152121?],
 ['o9_00374', 3.65724152121?],
 ['o9_00375', 3.657244227360?],
 ['o9_00376', 3.65731513751?],
 ['o9_00377', 3.65731513751?],
 ['o9_00378', 3.657320199915?],
 ['o9_00379', 3.657320199915?],
 ['o9_00380', 3.65735084793?],
 ['o9_00381', 3.65736030336?],
 ['o9_00382', 3.657422764758?],
 ['o9_00383', 3.65742276476?],
 ['o9_00384', 3.657437030518?],
 ['o9_00385', 3.657442039473?],
 ['o9_00386', 3.65747544938?],
 ['o9_00387', 3.657475449376?],
 ['o9_00388', 3.657502795756?],
 ['o9_00389', 3.657508906964?],
 ['o9_00390', 3.657551186181?],
 ['o9_00391', 3.657561050296?],
 ['o9_00392', 3.657573507350?],
 ['o9_00393', 3.657573507350?],
 ['o9_00394', 3.65758291635?],
 ['o9_00395', 3.65758291635?],
 ['o9_00396', 3.657614844922?],
 ['o9_00397', 3.657614844922?],
 ['t00318', 3.657677503743?],
 ['t00319', 3.65768515829?],
 ['o9_00398', 3.657700984511?],
 ['o9_00399', 3.657700984512?],
 ['o9_00400', 3.657704586030?],
 ['o9_00401', 3.657704586030?],
 ['o9_00402', 3.657758294451?],
 ['o9_00403', 3.657764344857?],
 ['o9_00404', 3.657814662430?],
 ['o9_00405', 3.65781466243?],
 ['o9_00406', 3.657827099351?],
 ['o9_00407', 3.65782709936?],
 ['o9_00408', 3.65783947111?],
 ['o9_00409', 3.65783947111?],
 ['o9_00410', 3.65787375013?],
 ['o9_00411', 3.657877962639?],
 ['o9_00412', 3.65807467712?],
 ['o9_00413', 3.65807467712?],
 ['o9_00414', 3.658083171050?],
 ['o9_00415', 3.658089694253?],
 ['o9_00416', 3.658106504672?],
 ['o9_00417', 3.658106504672?],
 ['o9_00418', 3.658110414203?],
 ['o9_00419', 3.658118519495?],
 ['o9_00420', 3.658124610029?],
 ['o9_00421', 3.658131829425?],
 ['o9_00422', 3.65814981135?],
 ['o9_00423', 3.65814981135?],
 ['o9_00424', 3.65816965529?],
 ['o9_00425', 3.65816965529?],
 ['o9_00426', 3.658195898841?],
 ['o9_00427', 3.658195898841?],
 ['o9_00428', 3.658200284743?],
 ['o9_00429', 3.658200284743?],
 ['o9_00430', 3.65822836251?],
 ['o9_00431', 3.658228362508?],
 ['o9_00432', 3.65823727204?],
 ['o9_00433', 3.65823727204?],
 ['o9_00434', 3.658259315257?],
 ['o9_00435', 3.658259315257?],
 ['o9_00436', 3.658267594174?],
 ['o9_00437', 3.658271156848?],
 ['o9_00438', 3.658304451420?],
 ['o9_00439', 3.658312162493?],
 ['o9_00440', 3.658315274000?],
 ['o9_00441', 3.658315274000?],
 ['o9_00442', 3.65831752980?],
 ['o9_00443', 3.658321906173?],
 ['o9_00444', 3.658382723223?],
 ['o9_00445', 3.658389216828?],
 ['o9_00446', 3.658405754081?],
 ['o9_00447', 3.658413168346?],
 ['o9_00448', 3.65843271905?],
 ['o9_00449', 3.65843271905?],
 ['o9_00450', 3.658450793879?],
 ['o9_00451', 3.658451563888?],
 ['o9_00452', 3.658456377025?],
 ['o9_00453', 3.658458199172?],
 ['o9_00454', 3.65845983206?],
 ['o9_00455', 3.65845983206?],
 ['o9_00456', 3.658461476792?],
 ['o9_00457', 3.658461476792?],
 ['o9_00458', 3.658510955083?],
 ['o9_00459', 3.658510955083?],
 ['o9_00460', 3.658521174377?],
 ['o9_00461', 3.658521174377?],
 ['o9_00462', 3.658559318420?],
 ['o9_00463', 3.658559318420?],
 ['o9_00464', 3.658656823334?],
 ['o9_00465', 3.658656823334?],
 ['o9_00466', 3.658659002401?],
 ['o9_00467', 3.658660573890?],
 ['o9_00468', 3.658660573890?],
 ['o9_00469', 3.65866154912?],
 ['o9_00470', 3.65866154912?],
 ['o9_00471', 3.658664373716?],
 ['o9_00472', 3.65867508009?],
 ['o9_00473', 3.658675080084?],
 ['o9_00474', 3.658681079645?],
 ['o9_00475', 3.658687686881?],
 ['o9_00476', 3.658721640076?],
 ['o9_00477', 3.658725965372?],
 ['o9_00478', 3.65872671714?],
 ['o9_00479', 3.6587267172?],
 ['o9_00480', 3.658763924199?],
 ['o9_00481', 3.658770391746?],
 ['o9_00482', 3.65880890966?],
 ['o9_00483', 3.658812046185?],
 ['o9_00484', 3.658849282782?],
 ['o9_00485', 3.658855557704?],
 ['o9_00486', 3.65885571741?],
 ['o9_00487', 3.658855717402?],
 ['o9_00488', 3.65887968592?],
 ['o9_00489', 3.65887968592?],
 ['o9_00490', 3.65890705326?],
 ['o9_00491', 3.65890705326?],
 ['o9_00492', 3.658908000155?],
 ['o9_00493', 3.658913427807?],
 ['o9_00494', 3.658914198620?],
 ['o9_00495', 3.658914198620?],
 ['o9_00496', 3.658936725846?],
 ['o9_00497', 3.658936774191?],
 ['o9_00498', 3.658942092940?],
 ['o9_00499', 3.658942811433?],
 ['o9_00500', 3.65894492636?],
 ['o9_00501', 3.65894492636?],
 ['o9_00502', 3.658951575878?],
 ['o9_00503', 3.65895157588?],
 ['o9_00504', 3.65901448343?],
 ['o9_00505', 3.65901448343?],
 ['o9_00506', 3.65903355842?],
 ['o9_00507', 3.659037276092?],
 ['o9_00508', 3.65904496470?],
 ['o9_00509', 3.659049876699?],
 ['o9_00510', 3.65909434144?],
 ['o9_00511', 3.65909434144?],
 ['o9_00512', 3.65912728010?],
 ['o9_00513', 3.65912728010?],
 ['o9_00514', 3.65913949148?],
 ['o9_00515', 3.65913949148?],
 ['o9_00516', 3.659155725933?],
 ['o9_00517', 3.659158944598?],
 ['o9_00518', 3.659168837890?],
 ['o9_00519', 3.659168837890?],
 ['o9_00520', 3.659171595910?],
 ['o9_00521', 3.659176607160?],
 ['o9_00522', 3.65923142586?],
 ['o9_00523', 3.65923142586?],
 ['o9_00524', 3.659254486032?],
 ['o9_00525', 3.659254486032?],
 ['o9_00526', 3.659260724017?],
 ['o9_00527', 3.659264740317?],
 ['o9_00528', 3.659317857200?],
 ['o9_00529', 3.659321943300?],
 ['o9_00530', 3.659321943300?],
 ['o9_00531', 3.659323014018?],
 ['o9_00532', 3.65935934119?],
 ['o9_00533', 3.65935934119?],
 ['o9_00534', 3.659393891990?],
 ['o9_00535', 3.65939459938?],
 ['o9_00536', 3.659396915893?],
 ['o9_00537', 3.659398331036?],
 ['o9_00538', 3.659424227823?],
 ['o9_00539', 3.659429123509?],
 ['o9_00540', 3.65948703219?],
 ['o9_00541', 3.659491116884?],
 ['o9_00542', 3.659580099780?],
 ['o9_00543', 3.659583831503?],
 ['o9_00544', 3.65967286056?],
 ['o9_00545', 3.659676244864?],
 ['o9_00546', 3.659686907098?],
 ['o9_00547', 3.659686907098?],
 ['o9_00548', 3.659690893390?],
 ['o9_00549', 3.659690893390?],
 ['o9_00550', 3.659710436776?],
 ['o9_00551', 3.659714231148?],
 ['o9_00552', 3.659723752687?],
 ['o9_00553', 3.659728028847?],
 ['o9_00554', 3.65973937589?],
 ['o9_00555', 3.65973937589?],
 ['o9_00556', 3.659745064334?],
 ['o9_00557', 3.659745064334?],
 ['o9_00558', 3.659756617944?],
 ['o9_00559', 3.659760776986?],
 ['o9_00560', 3.65979284705?],
 ['o9_00561', 3.65979284705?],
 ['o9_00562', 3.659817355928?],
 ['o9_00563', 3.659819774167?],
 ['o9_00564', 3.659870994763?],
 ['o9_00565', 3.659874418380?],
 ['o9_00566', 3.659875893939?],
 ['o9_00567', 3.659879862948?],
 ['o9_00568', 3.659884919347?],
 ['o9_00569', 3.659884919347?],
 ['o9_00570', 3.65993842229?],
 ['o9_00571', 3.65993842229?],
 ['o9_00572', 3.65994727310?],
 ['o9_00573', 3.65994875665?],
 ['o9_00574', 3.65994875665?],
 ['o9_00575', 3.659949680339?],
 ['o9_00576', 3.659964147540?],
 ['o9_00577', 3.659965659703?],
 ['o9_00578', 3.659965659703?],
 ['o9_00579', 3.659967920482?],
 ['o9_00580', 3.66003687820?],
 ['o9_00581', 3.660038989036?],
 ['o9_00582', 3.66012374126?],
 ['o9_00583', 3.660123741260?],
 ['o9_00584', 3.66015723743?],
 ['o9_00585', 3.660160008512?],
 ['o9_00586', 3.66020659635?],
 ['o9_00587', 3.660209486782?],
 ['o9_00588', 3.660238972982?],
 ['o9_00589', 3.660242208575?],
 ['o9_00590', 3.66025936184?],
 ['o9_00591', 3.66025936184?],
 ['o9_00592', 3.660287480359?],
 ['o9_00593', 3.660290658974?],
 ['o9_00594', 3.660319484755?],
 ['o9_00595', 3.660319484755?],
 ['o9_00596', 3.660348725426?],
 ['o9_00597', 3.660350576499?],
 ['o9_00598', 3.66039172642?],
 ['o9_00599', 3.660394289318?],
 ['o9_00600', 3.66039434370?],
 ['o9_00601', 3.66039434370?],
 ['o9_00602', 3.660411252662?],
 ['o9_00603', 3.660414206617?],
 ['o9_00604', 3.66044652847?],
 ['o9_00605', 3.66044652847?],
 ['o9_00606', 3.66045424630?],
 ['o9_00607', 3.660456087095?],
 ['o9_00608', 3.660495305069?],
 ['o9_00609', 3.660495305069?],
 ['o9_00610', 3.660505092072?],
 ['o9_00611', 3.660507907050?],
 ['o9_00612', 3.660523292024?],
 ['o9_00613', 3.660525460544?],
 ['o9_00614', 3.66057222681?],
 ['o9_00615', 3.66057222681?],
 ['o9_00616', 3.660625129710?],
 ['o9_00617', 3.660627229915?],
 ['o9_00618', 3.66070083075?],
 ['o9_00619', 3.66070083075?],
 ['o9_00620', 3.660750165282?],
 ['o9_00621', 3.660750165282?],
 ['o9_00622', 3.660871369236?],
 ['o9_00623', 3.660873180447?],
 ['o9_00624', 3.660907164346?],
 ['o9_00625', 3.660909045935?],
 ['o9_00626', 3.660925679990?],
 ['o9_00627', 3.660927348765?],
 ['o9_00628', 3.660928431321?],
 ['o9_00629', 3.660929710718?],
 ['o9_00630', 3.660973049353?],
 ['o9_00631', 3.660973049353?],
 ['o9_00632', 3.661009698874?],
 ['o9_00633', 3.661011319239?],
 ['o9_00634', 3.661013857796?],
 ['o9_00635', 3.661015872048?],
 ['o9_00636', 3.661281244017?],
 ['o9_00637', 3.661281244017?],
 ['o9_00638', 3.661501095253?],
 ['o9_00639', 3.661502209893?]]
In [10]:
bound=FKP_bound(volW.center(),snappy.Manifold('s011').volume(verified=True).center())
len(W.short_slopes(length=bound*math.sqrt(W.cusp_areas()[0]))[0])
Out[10]:
1278
In [11]:
children_of_W=[]
bound=FKP_bound(volW.center(),snappy.Manifold('s011').volume(verified=True).center())
print('The FKP-bound is:',bound)
for s in W.short_slopes(length=bound*math.sqrt(W.cusp_areas()[0]))[0]:
    W.dehn_fill(s,0)
    vol=better_volume(W,index=500,try_hard=True)
    if vol==0:
        print(s,vol)
    if vol!=0:
        if vol.endpoints()[1]<snappy.Manifold('s011').volume(verified=True).endpoints()[0]:
            print(s,vol,W.identify())
            children_of_W.append(W.identify()[0].name())
The FKP-bound is: 36.6148858554503
(1, 0) 0
(-10, 1) 3.476173989239? [o9_00000(0,0)]
(-9, 1) 3.434540885903? [t00000(0,0)]
(-8, 1) 3.377597408232? [v0000(0,0)]
(-7, 1) 3.296902414327? [s000(0,0)]
(-6, 1) 3.177293278601? [m039(0,0)]
(-5, 1) 2.989120282930? [m023(0,0)]
(-4, 1) 2.666744783449? [m009(0,0)]
(-3, 1) 2.029883212820? [m004(0,0), 4_1(0,0), K2_1(0,0), K4a1(0,0), otet02_00001(0,0)]
(-2, 1) 0
(-1, 1) 0
(0, 1) 0
(1, 1) 0
(2, 1) 0
(3, 1) 2.029883212820? [m003(0,0), otet02_00000(0,0)]
(4, 1) 2.666744783449? [m010(0,0)]
(5, 1) 2.989120282930? [m022(0,0)]
(6, 1) 3.177293278601? [m040(0,0)]
(7, 1) 3.296902414327? [s001(0,0)]
(8, 1) 3.377597408232? [v0001(0,0)]
(9, 1) 3.434540885903? [t00001(0,0)]
(10, 1) 3.476173989239? [o9_00001(0,0)]
(-9, 2) 3.460128933251? [s005(0,0)]
(-7, 2) 3.356692845142? [m059(0,0)]
(-5, 2) 3.163963228883? [m032(0,0), 6_1(0,0), K4_1(0,0), K6a3(0,0)]
(-3, 2) 2.828122088331? [m015(0,0), 5_2(0,0), K3_2(0,0), K5a1(0,0)]
(-1, 2) 2.5689706009367? [m007(0,0)]
(1, 2) 2.568970600937? [m006(0,0)]
(3, 2) 2.828122088331? [m017(0,0)]
(5, 2) 3.163963228883? [m033(0,0)]
(7, 2) 3.356692845142? [m058(0,0)]
(9, 2) 3.460128933251? [s004(0,0)]
(-8, 3) 3.463688556153? [m080(0,0)]
(-7, 3) 3.427205246274? [m074(0,0), 8_1(0,0), K5_2(0,0), K8a11(0,0)]
(-5, 3) 3.331744231641? [m053(0,0), 7_2(0,0), K4_2(0,0), K7a4(0,0)]
(-4, 3) 3.275871643944? [m045(0,0)]
(-2, 3) 3.177293278601? [m035(0,0)]
(-1, 3) 3.148509826441? [m029(0,0)]
(1, 3) 3.148509826441? [m030(0,0)]
(2, 3) 3.177293278601? [m037(0,0)]
(4, 3) 3.275871643944? [m046(0,0)]
(5, 3) 3.331744231641? [m054(0,0)]
(7, 3) 3.427205246274? [m073(0,0)]
(8, 3) 3.463688556153? [m079(0,0)]
(-7, 4) 3.486660146295? [m094(0,0), 9_2(0,0), K5_3(0,0), K9a27(0,0)]
(-5, 4) 3.439592889349? [m077(0,0)]
(-3, 4) 3.394540517062? [m067(0,0)]
(-1, 4) 3.366729420471? [m062(0,0)]
(1, 4) 3.366729420471? [m061(0,0)]
(3, 4) 3.394540517062? [m066(0,0)]
(5, 4) 3.439592889349? [m076(0,0)]
(7, 4) 3.486660146295? [m093(0,0)]
(-4, 5) 3.493881599711? [s009(0,0)]
(-3, 5) 3.483898578333? [m090(0,0)]
(-2, 5) 3.476173989239? [m085(0,0)]
(-1, 5) 3.471287882780? [s007(0,0)]
(1, 5) 3.471287882780? [s006(0,0)]
(2, 5) 3.476173989239? [m084(0,0)]
(3, 5) 3.483898578333? [m089(0,0)]
(4, 5) 3.493881599711? [s008(0,0)]
In [12]:
children_of_P=[]
bound=FKP_bound(volW.center(),snappy.Manifold('s011').volume(verified=True).center())
print('The FKP-bound is:',bound)
for s in P.short_slopes(length=bound*math.sqrt(P.cusp_areas()[0]))[0]:
    P.dehn_fill(s,0)
    vol=better_volume(P,index=500,try_hard=True)
    if vol==0:
        print(s,vol)
    if vol!=0:
        if vol.endpoints()[1]<snappy.Manifold('s011').volume(verified=True).endpoints()[0]:
            print(s,vol,P.identify())
            children_of_P.append(P.identify()[0].name())
The FKP-bound is: 36.6148858554503
(1, 0) 0
(-7, 1) 3.470395731439? [v0003(0,0)]
(-6, 1) 3.404658962502? [s003(0,0)]
(-5, 1) 3.300217628536? [m049(0,0)]
(-4, 1) 3.121334773013? [m027(0,0)]
(-3, 1) 2.781833912396? [m011(0,0)]
(-2, 1) 2.0298832128193? [m003(0,0), otet02_00000(0,0)]
(-1, 1) 0
(0, 1) 0
(1, 1) 0
(2, 1) 0
(3, 1) 2.568970600937? [m006(0,0)]
(4, 1) 3.059338057779? [m026(0,0)]
(5, 1) 3.277062185134? [m047(0,0)]
(6, 1) 3.394556686696? [s002(0,0)]
(7, 1) 3.465465224753? [v0002(0,0)]
(-7, 2) 3.481970896073? [m087(0,0)]
(-5, 2) 3.337191720007? [m055(0,0)]
(-3, 2) 2.944106486677? [m019(0,0)]
(-1, 2) 0
(1, 2) 2.0298832128193? [m003(0,0), otet02_00000(0,0)]
(3, 2) 2.828122088331? [m016(0,0), K3_1(0,0), K12n242(0,0)]
(5, 2) 3.308241554731? [m052(0,0)]
(7, 2) 3.474402775553? [m083(0,0)]
(-7, 3) 3.4971332877818? [m096(0,0)]
(-5, 3) 3.3805053992016? [m064(0,0)]
(-4, 3) 3.2756765600244? [m044(0,0)]
(-2, 3) 2.8281220883308? [m016(0,0), K3_1(0,0), K12n242(0,0)]
(-1, 3) 2.568970600937? [m006(0,0)]
(1, 3) 2.7818339123961? [m011(0,0)]
(2, 3) 2.944106486677? [m019(0,0)]
(4, 3) 3.252908048472? [m043(0,0)]
(5, 3) 3.362093204427? [m060(0,0)]
(7, 3) 3.489870168028? [m095(0,0)]
(-5, 4) 3.4245035087710? [m072(0,0)]
(-3, 4) 3.252908048472? [m043(0,0)]
(-1, 4) 3.059338057779? [m026(0,0)]
(1, 4) 3.1213347730123? [m027(0,0)]
(3, 4) 3.275676560025? [m044(0,0)]
(5, 4) 3.417914837238? [m071(0,0), K5_1(0,0)]
(-4, 5) 3.4179148372375? [m071(0,0), K5_1(0,0)]
(-3, 5) 3.362093204427? [m060(0,0)]
(-2, 5) 3.3082415547304? [m052(0,0)]
(-1, 5) 3.2770621851340? [m047(0,0)]
(1, 5) 3.3002176285354? [m049(0,0)]
(2, 5) 3.337191720007? [m055(0,0)]
(3, 5) 3.380505399202? [m064(0,0)]
(4, 5) 3.424503508771? [m072(0,0)]
(6, 5) 3.499922930053? [s010(0,0)]
(-5, 6) 3.4999229300529? [s010(0,0)]
(-1, 6) 3.394556686696? [s002(0,0)]
(1, 6) 3.404658962502? [s003(0,0)]
(-3, 7) 3.489870168028? [m095(0,0)]
(-2, 7) 3.4744027755531? [m083(0,0)]
(-1, 7) 3.4654652247525? [v0002(0,0)]
(1, 7) 3.470395731439? [v0003(0,0)]
(2, 7) 3.481970896073? [m087(0,0)]
(3, 7) 3.497133287782? [m096(0,0)]
In [13]:
len(children_of_W)
Out[13]:
54
In [14]:
len(children_of_P)
Out[14]:
52
In [15]:
print(children_of_W)
['o9_00000', 't00000', 'v0000', 's000', 'm039', 'm023', 'm009', 'm004', 'm003', 'm010', 'm022', 'm040', 's001', 'v0001', 't00001', 'o9_00001', 's005', 'm059', 'm032', 'm015', 'm007', 'm006', 'm017', 'm033', 'm058', 's004', 'm080', 'm074', 'm053', 'm045', 'm035', 'm029', 'm030', 'm037', 'm046', 'm054', 'm073', 'm079', 'm094', 'm077', 'm067', 'm062', 'm061', 'm066', 'm076', 'm093', 's009', 'm090', 'm085', 's007', 's006', 'm084', 'm089', 's008']
In [16]:
print(children_of_P)
['v0003', 's003', 'm049', 'm027', 'm011', 'm003', 'm006', 'm026', 'm047', 's002', 'v0002', 'm087', 'm055', 'm019', 'm003', 'm016', 'm052', 'm083', 'm096', 'm064', 'm044', 'm016', 'm006', 'm011', 'm019', 'm043', 'm060', 'm095', 'm072', 'm043', 'm026', 'm027', 'm044', 'm071', 'm071', 'm060', 'm052', 'm047', 'm049', 'm055', 'm064', 'm072', 's010', 's010', 's002', 's003', 'm095', 'm083', 'v0002', 'v0003', 'm087', 'm096']
In [17]:
[x[0] for x in might_be_minimizer].index('s011')
Out[17]:
80
In [18]:
not_children=[]
for x in might_be_minimizer[0:81]:
    if x[0] not in children_of_P+children_of_W:
        print(x[0])
        not_children.append(x[0])
m034
m036
m038
m069
m070
m078
m081
m082
s011
In [19]:
len(not_children)
Out[19]:
9

This list of 9 manifolds have lens space surgeries but they are not obtained by surgery on P or W. We continue by showing that these give volumes of lens spaces that are smaller than the values realized by W and P.

For that we use again the FKP bound on the volume to reduce this to a finite check.

In [26]:
def FKP_bound(vol_surgery_link,lower_volume_bound):
    '''
    Returns the Futer-Kalfagianni-Purcell bound, that gives a lower bound on the normalized 
    length of a slope to ensure that a filling with such a slope on a link with volume = vol_surgery_link 
    has volume at least lower_volume_bound.
    '''
    return 2*3.14159/sqrt(1-(lower_volume_bound/vol_surgery_link)^(2/3))+0.1
In [23]:
for x in lens_spaces:
    for y in not_children:
        if y==x[-2]:
            print(x[0],'=',x[2],'; Volume bound: ',better_volume(snappy.Manifold(x[-2])))
m034(1, 0) = L(4,1) ; Volume bound:  3.166333321250?
m036(-1, 1) = L(21,8) ; Volume bound:  3.177293278601?
m036(1, 0) = L(3,1) ; Volume bound:  3.177293278601?
m038(1, 0) = L(3,1) ; Volume bound:  3.177293278601?
m069(0, 1) = L(17,3) ; Volume bound:  3.402991251167?
m069(1, 0) = L(12,5) ; Volume bound:  3.402991251167?
m069(1, 1) = L(29,8) ; Volume bound:  3.402991251167?
m070(0, 1) = L(23,5) ; Volume bound:  3.402991251167?
m070(1, 0) = L(4,1) ; Volume bound:  3.402991251167?
m070(1, 1) = L(27,8) ; Volume bound:  3.402991251167?
m078(1, 0) = L(5,1) ; Volume bound:  3.460675847482?
m081(1, 0) = L(11,2) ; Volume bound:  3.464408817290?
m081(1, 1) = L(5,1) ; Volume bound:  3.464408817290?
m082(0, 1) = L(27,8) ; Volume bound:  3.474247761313?
s011(0, 1) = L(38,7) ; Volume bound:  3.502349676519?
s011(1, 0) = L(9,1) ; Volume bound:  3.502349676519?
s011(1, 1) = L(29,9) ; Volume bound:  3.502349676519?
In [34]:
P=snappy.Manifold('m125')
W=snappy.Manifold('m129')
volP=better_volume(P)
volW=better_volume(W)
print(volW,volP)
3.663862376709? 3.663862376709?
In [35]:
bound=FKP_bound(volW.center(),snappy.Manifold('m034').volume(verified=True).center())
print('The FKP-bound is:',bound)
for s in W.short_slopes(length=bound*math.sqrt(W.cusp_areas()[0]))[0]:
    W.dehn_fill(s,0)
    vol=better_volume(W,index=500,try_hard=True)
    W.dehn_fill((1,0),1)
    if W.homology().order()==4:
        print(s,vol,W.identify())
    W.dehn_fill((0,0),1)
The FKP-bound is: 20.7352931115192
(-6, 1) 3.177293278601? []
(2, 1) 0 []
(-10, 3) 3.517760728543? []
(-2, 3) 3.177293278601? []
(-14, 5) 3.596882244932? []
(-6, 5) 3.517760728543? []
(-18, 7) 3.625796417488? []
(-10, 7) 3.596882244932? []
(-22, 9) 3.639386687137? []
(-14, 9) 3.625796417488? []
(-18, 11) 3.639386687137? []
In [36]:
bound=FKP_bound(volP.center(),snappy.Manifold('m034').volume(verified=True).center())
print('The FKP-bound is:',bound)
for s in P.short_slopes(length=bound*math.sqrt(P.cusp_areas()[0]))[0]:
    P.dehn_fill(s,0)
    vol=better_volume(P,index=500,try_hard=True)
    P.dehn_fill((1,0),1)
    if P.homology().order()==4:
        print(s,vol,P.identify())
    P.dehn_fill((0,0),1)
The FKP-bound is: 20.7352931115192
(0, 1) 0 []
(8, 5) 3.551403873987? []
(8, 7) 3.575972812121? []
(16, 11) 3.637604216941? []
(16, 13) 3.640592224262? []

This proves that vol(L(4,1)) is at most vol(m034)=3.166333321250. On the other hand, we have seen that any surgery on W or P that yields a hyperbolic manifold with first homology of order 4 has larger volume. And thus the volume of L(4,1) is not realized on W and P.

Similarly, this can be done for other manifolds from the above list.