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.
import snappy
import csv
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)
205823
lens_spaces=[]
for x in exceptional_fillings:
if x[1]=='lens_space':
lens_spaces.append(x)
len(lens_spaces)
44487
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)
38056
This is a list of census knots with lens space surgeries. We describe the parent diagram of W and P.
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
P=snappy.Manifold('m125')
W=snappy.Manifold('m129')
volP=better_volume(P)
volW=better_volume(W)
print(volW,volP)
3.663862376709? 3.663862376709?
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])
might_be_minimizer.sort(key=lambda x: x[1])
len(might_be_minimizer)
1276
might_be_minimizer
[['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?], ['v0042', 3.610225138482?], ['s069', 3.610629872967?], ['v0043', 3.610882492798?], ['v0044', 3.610964594218?], ['v0045', 3.610964594218?], ['s070', 3.611333673853?], ['s071', 3.611333673853?], ['s072', 3.612383987822?], ['s073', 3.612674238432?], ['s074', 3.612949127828?], ['v0046', 3.61300646506?], ['v0047', 3.613006465061?], ['s075', 3.613316904965?], ['o9_00014', 3.61421285440?], ['o9_00015', 3.61421285440?], ['v0048', 3.614796030792?], ['v0049', 3.61503109011?], ['t00028', 3.615412876468?], ['v0050', 3.615567529849?], ['v0051', 3.615760355871?], ['v0052', 3.615760355871?], ['t00029', 3.61598657633?], ['v0053', 3.61599606731?], ['o9_00016', 3.61679304741?], ['v0054', 3.61722871469?], ['v0055', 3.61722871469?], ['s076', 3.617251533833?], ['s077', 3.617251533834?], ['t00030', 3.617414381437?], ['t00031', 3.617414381437?], ['t00032', 3.61814068845?], ['t00033', 3.618140688441?], ['t00034', 3.618316603412?], ['o9_00018', 3.61865713209?], ['o9_00019', 3.61865713209?], ['t00035', 3.618790392429?], ['v0056', 3.618973982147?], ['v0057', 3.619409278094?], ['v0058', 3.619409278095?], ['v0059', 3.619461392598?], ['v0060', 3.619936619027?], ['v0061', 3.61993661903?], ['v0062', 3.620037229293?], ['v0063', 3.620037229293?], ['o9_00020', 3.620715577913?], ['o9_00021', 3.62094746434?], ['t00036', 3.620990117254?], ['t00037', 3.620990117254?], ['s078', 3.621276553808?], ['s079', 3.621640464426?], ['o9_00023', 3.62268440821?], ['v0064', 3.622764179925?], ['v0065', 3.623034417344?], ['v0066', 3.623034417344?], ['v0067', 3.62312379403?], ['t00038', 3.62353997314?], ['t00039', 3.623539973140?], ['v0068', 3.623715978612?], ['v0069', 3.623715978612?], ['v0070', 3.62393743569?], ['v0071', 3.623937435683?], ['v0072', 3.624205696237?], ['v0073', 3.624464946982?], ['v0074', 3.624481536022?], ['v0075', 3.62448153603?], ['o9_00024', 3.6246947093?], ['o9_00025', 3.62469470925?], ['o9_00026', 3.62479023273?], ['o9_00027', 3.62479023273?], ['t00040', 3.625160673974?], ['t00041', 3.625160673974?], ['v0076', 3.625796417488?], ['v0077', 3.62579641749?], ['v0078', 3.62579641749?], ['v0079', 3.625796417487?], ['t00042', 3.625960933716?], ['t00043', 3.625960933716?], ['t00044', 3.626545341382?], ['t00045', 3.62677736198?], ['v0080', 3.62686672058?], ['v0081', 3.626866720577?], ['v0082', 3.627040079403?], ['v0083', 3.627208689698?], ['o9_00028', 3.62735368675?], ['o9_00029', 3.62735368675?], ['o9_00030', 3.627671214122?], ['v0084', 3.627861429328?], ['v0085', 3.627861429327?], ['o9_00031', 3.627991279470?], ['v0086', 3.628076385622?], ['v0087', 3.628317977696?], ['v0088', 3.628337430912?], ['v0089', 3.628337430912?], ['t00046', 3.628612639801?], ['t00047', 3.62890018457?], ['t00048', 3.628928096955?], ['t00049', 3.62892809696?], ['v0090', 3.629104156456?], ['v0091', 3.629104156456?], ['o9_00032', 3.62955678729?], ['v0092', 3.62973489932?], ['v0093', 3.629734899316?], ['t00050', 3.629736746371?], ['o9_00033', 3.629827741454?], ['t00051', 3.62983227267?], ['t00052', 3.63004999873?], ['t00053', 3.63004999873?], ['s080', 3.63023796976?], ['o9_00034', 3.63059701824?], ['o9_00035', 3.63059701824?], ['o9_00036', 3.6306728240?], ['o9_00037', 3.63067282392?], ['t00054', 3.630690633403?], ['t00055', 3.63086551552?], ['v0094', 3.631079691083?], ['v0095', 3.631079691083?], ['v0096', 3.631368047770?], ['t00056', 3.631425250641?], ['v0097', 3.63162925440?], ['t00057', 3.63163285400?], ['v0098', 3.63178943633?], ['v0099', 3.631789436327?], ['v0100', 3.632314996117?], ['v0101', 3.632314996117?], ['v0102', 3.63233504934?], ['t00058', 3.63247519542?], ['t00059', 3.632475195421?], ['v0103', 3.63255893644?], ['v0104', 3.632636382035?], ['v0105', 3.632748806855?], ['t00060', 3.633389388141?], ['t00061', 3.633593338153?], ['t00062', 3.633593338153?], ['t00063', 3.633614229312?], ['v0106', 3.634463257048?], ['o9_00038', 3.63450770191?], ['o9_00039', 3.63450770191?], ['v0107', 3.634555397686?], ['v0108', 3.634654431115?], ['o9_00040', 3.63466211457?], ['o9_00041', 3.63466211457?], ['v0109', 3.634768544378?], ['v0110', 3.634789162682?], ['v0111', 3.634789162682?], ['o9_00042', 3.63490078747?], ['o9_00043', 3.63490078747?], ['t00064', 3.635107833245?], ['t00065', 3.635107833245?], ['v0112', 3.635148198679?], ['v0113', 3.635148198679?], ['v0114', 3.635251186672?], ['v0115', 3.63536871377?], ['t00066', 3.63539295119?], ['t00067', 3.635392951183?], ['v0116', 3.635484565396?], ['v0117', 3.635484565396?], ['v0118', 3.635510000692?], ['v0119', 3.63568591356?], ['v0120', 3.635751612691?], ['v0121', 3.635751612691?], ['v0122', 3.635974086883?], ['v0123', 3.635974086883?], ['o9_00044', 3.636178290307?], ['o9_00045', 3.636178290307?], ['t00068', 3.63653743878?], ['t00069', 3.636537438774?], ['t00070', 3.636600267258?], ['t00071', 3.63660026726?], ['o9_00046', 3.63666665287?], ['o9_00047', 3.63666665287?], ['t00072', 3.637139651600?], ['t00073', 3.637139651600?], ['o9_00048', 3.637160192357?], ['v0124', 3.6372097779762?], ['v0125', 3.637209777976?], ['o9_00049', 3.63726371746?], ['v0126', 3.637317227409?], ['v0127', 3.637491469245?], ['t00074', 3.637604216940?], ['t00075', 3.637639443419?], ['t00076', 3.637719525468?], ['t00077', 3.63776513065?], ['t00078', 3.637800369805?], ['t00079', 3.637800369805?], ['t00080', 3.638237558728?], ['t00081', 3.63826548292?], ['t00082', 3.638265482918?], ['t00083', 3.63827898606?], ['t00084', 3.63827898606?], ['t00085', 3.63837700385?], ['v0128', 3.638407633913?], ['t00086', 3.63843097097?], ['t00087', 3.63843097097?], ['v0129', 3.63854492459?], ['o9_00050', 3.638765655630?], ['o9_00051', 3.638810144405?], ['o9_00052', 3.638937912711?], ['t00088', 3.639017420780?], ['t00089', 3.639017420780?], ['o9_00053', 3.63908279324?], ['o9_00054', 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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?]]
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])
1278
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)]
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)]
len(children_of_W)
54
len(children_of_P)
52
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']
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']
[x[0] for x in might_be_minimizer].index('s011')
80
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
len(not_children)
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.
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
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?
P=snappy.Manifold('m125')
W=snappy.Manifold('m129')
volP=better_volume(P)
volW=better_volume(W)
print(volW,volP)
3.663862376709? 3.663862376709?
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? []
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.