In this notebook we compute the formal semigroups (as defined by S. Wang [Semigroups of L-space knots and nonalgebraic iterated torus knots]) of the SnapPy census L-space knots and check weather they are semigroups or not.

import snappy

def formal_semigroup(alex):
    '''Takes the Alexander polynomial of an L-space knot and returns its formal semigroup.'''
    for x in alex.coefficients():
        if abs(x)!=1:
            raise ValueError('The Alexander polynomial is not the polynomial of an L-space.')
    fsg=[]
    exp=alex.exponents()
    g=alex.degree()
    for i in range((len(exp)-1)/2):
        x=exp[2*i]
        y=exp[2*i+1]
        for j in range(g):
            if x+j<y:
                fsg.append(x+j)
    fsg.append(g)
    return [fsg,'+all values greater than',g]

def is_semigroup(fsg):
    '''Takes a formal semigroup and checks if it is a semi group.'''
    for x in fsg[0]:
        for y in fsg[0]:
            if x+y<=fsg[2]:
                if x+y not in fsg[0]:
                    return False # If x+y is not in fsg it is NOT a semigroup.
    return True # If this is the case it is a semigroup.

# The list of census L-space knots.
L_space_knots=['m016', 'm071', 'm082', 'm103', 'm118', 'm144', 'm194', 'm198', 'm211', 'm223', 'm239', 'm240', 'm270', 'm276', 'm281', 's042', 's068', 's086', 's104', 's114', 's294', 's301', 's308', 's336', 's344', 's346', 's367', 's369', 's384', 's407', 's560', 's582', 's652', 's665', 's682', 's684', 's769', 's800', 's849', 'v0082', 'v0114', 'v0165', 'v0220', 'v0223', 'v0249', 'v0319', 'v0330', 'v0398', 'v0407', 'v0424', 'v0434', 'v0497', 'v0545', 'v0554', 'v0570', 'v0573', 'v0707', 'v0709', 'v0715', 'v0740', 'v0741', 'v0759', 'v0765', 'v0830', 'v0847', 'v0912', 'v0939', 'v0945', 'v0959', 'v1077', 'v1109', 'v1269', 'v1300', 'v1359', 'v1392', 'v1423', 'v1425', 'v1547', 'v1565', 'v1620', 'v1628', 'v1690', 'v1709', 'v1716', 'v1718', 'v1728', 'v1810', 'v1832', 'v1839', 'v1915', 'v1921', 'v1940', 'v1966', 'v1980', 'v1986', 'v2024', 'v2090', 'v2215', 'v2217', 'v2290', 'v2325', 'v2384', 'v2759', 'v2871', 'v2900', 'v2925', 'v2930', 'v3070', 'v3105', 'v3234', 'v3335', 'v3354', 'v3482', 't00110', 't00146', 't00324', 't00423', 't00434', 't00550', 't00621', 't00729', 't00787', 't00826', 't00855', 't00873', 't00932', 't01033', 't01037', 't01125', 't01216', 't01268', 't01292', 't01318', 't01368', 't01409', 't01422', 't01424', 't01440', 't01598', 't01636', 't01646', 't01690', 't01757', 't01815', 't01834', 't01850', 't01863', 't01949', 't01966', 't02099', 't02104', 't02238', 't02276', 't02378', 't02398', 't02404', 't02470', 't02537', 't02567', 't02639', 't03106', 't03566', 't03607', 't03709', 't03710', 't03713', 't03781', 't03843', 't03864', 't03956', 't03979', 't04003', 't04019', 't04102', 't04180', 't04228', 't04244', 't04382', 't04449', 't04557', 't04721', 't04756', 't04927', 't05118', 't05239', 't05390', 't05425', 't05426', 't05538', 't05564', 't05578', 't05658', 't05663', 't05674', 't05695', 't06001', 't06246', 't06440', 't06463', 't06525', 't06570', 't06573', 't06605', 't06637', 't06715', 't06957', 't07070', 't07104', 't07348', 't07355', 't07412', 't07670', 't08111', 't08114', 't08184', 't08201', 't08267', 't08273', 't08403', 't08532', 't08576', 't08936', 't09016', 't09126', 't09267', 't09284', 't09313', 't09450', 't09455', 't09500', 't09580', 't09633', 't09690', 't09704', 't09847', 't09852', 't09882', 't09912', 't09954', 't10177', 't10188', 't10215', 't10224', 't10230', 't10262', 't10292', 't10462', 't10496', 't10643', 't10681', 't10832', 't10985', 't11198', 't11376', 't11548', 't11556', 't11852', 't11887', 't11909', 't12288', 't12533', 't12681', 't12753', 'o9_00133', 'o9_00168', 'o9_00644', 'o9_00797', 'o9_00815', 'o9_01079', 'o9_01175', 'o9_01436', 'o9_01496', 'o9_01584', 'o9_01621', 'o9_01680', 'o9_01765', 'o9_01936', 'o9_01953', 'o9_01955', 'o9_02255', 'o9_02340', 'o9_02350', 'o9_02383', 'o9_02386', 'o9_02655', 'o9_02696', 'o9_02706', 'o9_02735', 'o9_02772', 'o9_02786', 'o9_02794', 'o9_02909', 'o9_03032', 'o9_03108', 'o9_03118', 'o9_03133', 'o9_03149', 'o9_03162', 'o9_03188', 'o9_03288', 'o9_03313', 'o9_03412', 'o9_03526', 'o9_03586', 'o9_03622', 'o9_03802', 'o9_03833', 'o9_03932', 'o9_04054', 'o9_04060', 'o9_04106', 'o9_04205', 'o9_04245', 'o9_04269', 'o9_04313', 'o9_04431', 'o9_04435', 'o9_04438', 'o9_04938', 'o9_05021', 'o9_05177', 'o9_05229', 'o9_05287', 'o9_05357', 'o9_05426', 'o9_05483', 'o9_05562', 'o9_05618', 'o9_05860', 'o9_05970', 'o9_06060', 'o9_06128', 'o9_06154', 'o9_06248', 'o9_06301', 'o9_06956', 'o9_07044', 'o9_07152', 'o9_07167', 'o9_07195', 'o9_07401', 'o9_07790', 'o9_07893', 'o9_07943', 'o9_07945', 'o9_08006', 'o9_08042', 'o9_08224', 'o9_08302', 'o9_08402', 'o9_08477', 'o9_08497', 'o9_08647', 'o9_08765', 'o9_08771', 'o9_08776', 'o9_08828', 'o9_08831', 'o9_08852', 'o9_08875', 'o9_09052', 'o9_09213', 'o9_09271', 'o9_09372', 'o9_09465', 'o9_09731', 'o9_09808', 'o9_10020', 'o9_10192', 'o9_10213', 'o9_10696', 'o9_11002', 'o9_11100', 'o9_11248', 'o9_11467', 'o9_11537', 'o9_11541', 'o9_11556', 'o9_11560', 'o9_11570', 'o9_11658', 'o9_11685', 'o9_11795', 'o9_11845', 'o9_11999', 'o9_12079', 'o9_12144', 'o9_12230', 'o9_12253', 'o9_12412', 'o9_12459', 'o9_12477', 'o9_12519', 'o9_12693', 'o9_12736', 'o9_12757', 'o9_12873', 'o9_12892', 'o9_12919', 'o9_12971', 'o9_13052', 'o9_13054', 'o9_13056', 'o9_13125', 'o9_13182', 'o9_13188', 'o9_13400', 'o9_13403', 'o9_13433', 'o9_13508', 'o9_13537', 'o9_13604', 'o9_13639', 'o9_13649', 'o9_13666', 'o9_13720', 'o9_13952', 'o9_14018', 'o9_14079', 'o9_14108', 'o9_14136', 'o9_14359', 'o9_14364', 'o9_14376', 'o9_14495', 'o9_14599', 'o9_14716', 'o9_14831', 'o9_14974', 'o9_15506', 'o9_15633', 'o9_15808', 'o9_15997', 'o9_16065', 'o9_16141', 'o9_16157', 'o9_16181', 'o9_16319', 'o9_16356', 'o9_16431', 'o9_16514', 'o9_16527', 'o9_16642', 'o9_16685', 'o9_16748', 'o9_16920', 'o9_17382', 'o9_17450', 'o9_17646', 'o9_18007', 'o9_18209', 'o9_18341', 'o9_18633', 'o9_18646', 'o9_18813', 'o9_19130', 'o9_19247', 'o9_19364', 'o9_19396', 'o9_19645', 'o9_19724', 'o9_20029', 'o9_20219', 'o9_20305', 'o9_20364', 'o9_20472', 'o9_21195', 'o9_21496', 'o9_21513', 'o9_21620', 'o9_21893', 'o9_21918', 'o9_22129', 'o9_22252', 'o9_22477', 'o9_22607', 'o9_22663', 'o9_22698', 'o9_22925', 'o9_23023', 'o9_23032', 'o9_23179', 'o9_23263', 'o9_23461', 'o9_23660', 'o9_23723', 'o9_23955', 'o9_23961', 'o9_23971', 'o9_23977', 'o9_24069', 'o9_24126', 'o9_24149', 'o9_24183', 'o9_24290', 'o9_24401', 'o9_24407', 'o9_24534', 'o9_24592', 'o9_24779', 'o9_24886', 'o9_24889', 'o9_24946', 'o9_25110', 'o9_25199', 'o9_25341', 'o9_25444', 'o9_25595', 'o9_25709', 'o9_25832', 'o9_26141', 'o9_26471', 'o9_26570', 'o9_26604', 'o9_26767', 'o9_26791', 'o9_27107', 'o9_27155', 'o9_27261', 'o9_27371', 'o9_27392', 'o9_27429', 'o9_27480', 'o9_27737', 'o9_27767', 'o9_28113', 'o9_28153', 'o9_28284', 'o9_28529', 'o9_28592', 'o9_28746', 'o9_28751', 'o9_28810', 'o9_29048', 'o9_29246', 'o9_29436', 'o9_29529', 'o9_29551', 'o9_29648', 'o9_29751', 'o9_29766', 'o9_30142', 'o9_30150', 'o9_30375', 'o9_30634', 'o9_30650', 'o9_30721', 'o9_30790', 'o9_31165', 'o9_31267', 'o9_31321', 'o9_31440', 'o9_31481', 'o9_32044', 'o9_32065', 'o9_32132', 'o9_32150', 'o9_32257', 'o9_32314', 'o9_32471', 'o9_32588', 'o9_32964', 'o9_33189', 'o9_33284', 'o9_33380', 'o9_33430', 'o9_33486', 'o9_33526', 'o9_33585', 'o9_33801', 'o9_33944', 'o9_33959', 'o9_34000', 'o9_34403', 'o9_34409', 'o9_34689', 'o9_35320', 'o9_35549', 'o9_35666', 'o9_35682', 'o9_35720', 'o9_35736', 'o9_35772', 'o9_35928', 'o9_36114', 'o9_36250', 'o9_36380', 'o9_36544', 'o9_36809', 'o9_36958', 'o9_37050', 'o9_37291', 'o9_37482', 'o9_37551', 'o9_37685', 'o9_37751', 'o9_37754', 'o9_37851', 'o9_37941', 'o9_38287', 'o9_38679', 'o9_38811', 'o9_38928', 'o9_38989', 'o9_39162', 'o9_39394', 'o9_39451', 'o9_39521', 'o9_39606', 'o9_39608', 'o9_39859', 'o9_39879', 'o9_39981', 'o9_40026', 'o9_40052', 'o9_40075', 'o9_40179', 'o9_40363', 'o9_40487', 'o9_40504', 'o9_40582', 'o9_41372', 'o9_42224', 'o9_42493', 'o9_42675', 'o9_42961', 'o9_43001', 'o9_43679', 'o9_43750', 'o9_43857', 'o9_43953', 'o9_44054']

len(L_space_knots)

formal_semigroups=[]
for name in L_space_knots:
    K=snappy.Manifold(name)
    alex=K.alexander_polynomial()
    fsg=formal_semigroup(alex)
    formal_semigroups.append([name,is_semigroup(fsg),fsg])

formal_semigroups

for x in formal_semigroups:
    if x[1]==True:
        print(x)

We have shown that o9_30634 and t09847 are the only census L-space knots whose formal semigroup is a semigroup. In particular, both are negative answers to Question 2.8 in Wang's paper. It is known that any algebraic knot is an L-space knot whose formal semigroup is a semigroup. On the other hand, algebraic knots are known to be iterated cables of torus knots and are therefore not hyperbolic. Question 2.8 in Wang's paper asks if there exists an L-space knot which is not an iterated torus knot whose formal semigroup is a semigroup. Since our examples are hyperbolic they cannot be iterated torus knots and thus negatively answers Wang's question. Actually, both knots fit into an infinite family of hyperbolic L-space knots whose formal semigroups are semigroups.

We work with the infinite family that corresponds to o9_30634. The other family can be studied in exactly the same way.

for n in range(1,30):
    word=(2*n+1)*[2,1,3,2]+[-1,2,1,1,2] # n=1 yields the knot 09_30634.
    D=snappy.Link(braid_closure=word)
    K=D.exterior()
    alex=K.alexander_polynomial()
    fsg=formal_semigroup(alex)
    print(2*n+1,is_semigroup(fsg),fsg)

#The other infinity family where n=1 corresponds to t09847. The family can be handled exactly as the other example of the infinite family in our paper.

for n in range(1,30):
    word=(2*n+1)*[2,1,3,2]+[1,2,1,1,2]
    D=snappy.Link(braid_closure=word)
    K=D.exterior()
    alex=K.alexander_polynomial()
    fsg=formal_semigroup(alex)
    print(2*n+1,is_semigroup(fsg),fsg)