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Flat knot 5.88

Min(phi) over symmetries of the knot is: [-2,0,2,0,3,1]
Flat knots (up to 7 crossings) with same phi are :['5.88', '7.727', '7.29503', '7.30058', '7.35738', '7.38629']
Arrow polynomial of the knot is: 4*K1**2 + 4*K1*K2 - 2*K1 - 2*K2 - 2*K3 - 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.16', '5.35', '5.81', '5.88', '7.948', '7.1722', '7.1729', '7.5258', '7.5895', '7.5945', '7.6163', '7.6518', '7.8144', '7.9039', '7.9082', '7.9287', '7.9765', '7.9910', '7.10238', '7.10252', '7.10254', '7.10256', '7.10422', '7.10630', '7.10752', '7.10798', '7.10825', '7.11020', '7.11077', '7.11264', '7.11266', '7.11402', '7.11477', '7.11479', '7.11874', '7.12092', '7.12257', '7.12265', '7.12502', '7.12918', '7.12971', '7.13609', '7.13914', '7.13956', '7.14059', '7.14149', '7.14198', '7.14685', '7.14779', '7.15082', '7.15338', '7.15344', '7.15664', '7.15670', '7.15681', '7.15717', '7.16041', '7.16075', '7.16139', '7.16189', '7.17447', '7.17660', '7.17773', '7.17799', '7.17822', '7.17826', '7.18033', '7.18051', '7.18126', '7.18147', '7.18151', '7.18161', '7.18165', '7.18375', '7.18388', '7.18396', '7.18475', '7.18479', '7.18511', '7.18515', '7.18566', '7.18570', '7.18689', '7.18690', '7.18703', '7.18704', '7.18928', '7.18930', '7.19011', '7.19012', '7.19049', '7.19050', '7.19103', '7.19104', '7.19347', '7.19370', '7.19417', '7.19818', '7.19924', '7.19938', '7.20224', '7.20322', '7.20335', '7.20363', '7.20432', '7.20944', '7.21365', '7.21461', '7.21680', '7.21786', '7.22374', '7.22378', '7.22640', '7.22752', '7.22849', '7.22862', '7.23269', '7.23343', '7.23410', '7.23428', '7.23552', '7.23681', '7.23761', '7.23882', '7.24099', '7.24347', '7.24933', '7.25172', '7.25174', '7.25298', '7.25300', '7.25340', '7.25499', '7.25720', '7.25848', '7.25916', '7.25919', '7.25986', '7.25988', '7.26005', '7.26017', '7.26185', '7.26288', '7.26358', '7.26649', '7.26658', '7.26670', '7.26773', '7.26782', '7.26792', '7.26823', '7.26835', '7.27020', '7.27021', '7.27025', '7.27029', '7.27030', '7.27092', '7.27113', '7.27123', '7.27132', '7.27228', '7.27235', '7.27245', '7.27260', '7.27304', '7.27340', '7.27346', '7.27401', '7.27577', '7.27584', '7.27661', '7.27666', '7.27672', '7.27688', '7.27887', '7.27900', '7.27951', '7.28230', '7.28364', '7.28395', '7.28513', '7.28549', '7.28598', '7.28714', '7.28770', '7.28858', '7.29001', '7.29059', '7.29100', '7.29847', '7.30114', '7.30137', '7.30441', '7.30563', '7.30616', '7.30638', '7.30706', '7.30963', '7.31013', '7.31024', '7.31206', '7.31209', '7.31335', '7.31463', '7.31664', '7.32084', '7.32592', '7.32806', '7.32814', '7.33295', '7.33505', '7.33619', '7.33898', '7.34106', '7.34586', '7.36057', '7.36058', '7.36158', '7.36159', '7.36806', '7.36890', '7.36991', '7.37007', '7.37010', '7.37122', '7.37125', '7.37233', '7.37294', '7.37300', '7.37336', '7.37386', '7.37481', '7.37640', '7.37644', '7.37702', '7.37840', '7.37863', '7.37868', '7.37953', '7.38182', '7.38187', '7.38232', '7.38445', '7.39566', '7.39573', '7.39613', '7.39641', '7.39831', '7.39907', '7.39908', '7.39930', '7.40006', '7.40023', '7.40047', '7.40141', '7.40162', '7.40170', '7.40201', '7.40230', '7.40239', '7.40282', '7.40283', '7.40300', '7.40353', '7.40487', '7.40492', '7.40686', '7.41099', '7.41229', '7.41268', '7.41270', '7.41474', '7.41507', '7.41598', '7.41601', '7.42471', '7.42480', '7.42486', '7.42512', '7.42607', '7.42633', '7.44210']
Outer characteristic polynomial of the knot is: t^4+18t^2+4
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.88', '7.29503', '7.38629']
2-strand cable arrow polynomial of the knot is: 128*K1**4*K2 - 480*K1**4 - 224*K1**2*K2**2 + 648*K1**2*K2 - 96*K1**2*K3**2 - 128*K1**2*K4**2 - 596*K1**2 + 656*K1*K2*K3 + 408*K1*K3*K4 + 152*K1*K4*K5 - 16*K2**4 - 48*K2**2*K3**2 - 48*K2**2*K4**2 + 144*K2**2*K4 - 572*K2**2 + 56*K2*K3*K5 + 32*K2*K4*K6 - 416*K3**2 - 252*K4**2 - 60*K5**2 - 4*K6**2 + 714
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.88']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.142', 'vk5.153', 'vk5.157', 'vk5.266', 'vk5.270', 'vk5.281', 'vk5.285', 'vk5.571', 'vk5.572', 'vk5.1043', 'vk5.1059', 'vk5.1357', 'vk5.1368', 'vk5.1372', 'vk5.1579', 'vk5.1580']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3U2U4O5O4U1U5U3
R3 orbit {'O1O2O3U2U4O5O4U1U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U3O5O4U5U2
Gauss code of K* O1O2O3U1U4U3O4O5U2U5
Gauss code of -K* O1O2O3U4U2O4O5U1U5U3
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -1 2 1 0],[ 2 0 0 3 2 0],[ 1 0 0 1 1 0],[-2 -3 -1 0 -1 -1],[-1 -2 -1 1 0 0],[ 0 0 0 1 0 0]]
Primitive based matrix [[ 0 2 0 -2],[-2 0 -1 -3],[ 0 1 0 0],[ 2 3 0 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,0,2,1,3,0]
Phi over symmetry [-2,0,2,0,3,1]
Phi of -K [-2,0,2,2,1,1]
Phi of K* [-2,0,2,1,1,2]
Phi of -K* [-2,0,2,0,3,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial -9z-17
Enhanced Jones-Krushkal polynomial -9w^2z-17w
Inner characteristic polynomial t^3+10t
Outer characteristic polynomial t^4+18t^2+4
Flat arrow polynomial 4*K1**2 + 4*K1*K2 - 2*K1 - 2*K2 - 2*K3 - 1
2-strand cable arrow polynomial 128*K1**4*K2 - 480*K1**4 - 224*K1**2*K2**2 + 648*K1**2*K2 - 96*K1**2*K3**2 - 128*K1**2*K4**2 - 596*K1**2 + 656*K1*K2*K3 + 408*K1*K3*K4 + 152*K1*K4*K5 - 16*K2**4 - 48*K2**2*K3**2 - 48*K2**2*K4**2 + 144*K2**2*K4 - 572*K2**2 + 56*K2*K3*K5 + 32*K2*K4*K6 - 416*K3**2 - 252*K4**2 - 60*K5**2 - 4*K6**2 + 714
Genus of based matrix 1
Fillings of based matrix [[{1, 5}, {2, 4}, {3}], [{2, 5}, {1, 4}, {3}], [{3, 5}, {2, 4}, {1}], [{5}, {2, 4}, {1, 3}]]
If K is slice False
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