Table of flat knot invariants
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Flat knot 5.82

Min(phi) over symmetries of the knot is: [-2,-1,1,1,1,0,0,1,2,1,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['5.82', '6.1734', '7.13522', '7.32803']
Arrow polynomial of the knot is: -4*K1**3 + 6*K1**2 + 4*K1*K2 + K1 - 3*K2 - K3 - 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.53', '5.82', '7.1849', '7.2671', '7.3671', '7.4969', '7.5203', '7.5310', '7.5640', '7.6447', '7.6649', '7.6741', '7.6775', '7.6796', '7.6860', '7.7483', '7.7719', '7.7775', '7.7843', '7.7946', '7.7999', '7.8438', '7.8453', '7.8473', '7.8813', '7.8952', '7.9734', '7.9989', '7.11734', '7.11741', '7.11972', '7.12043', '7.12045', '7.12073', '7.12095', '7.12227', '7.12238', '7.12321', '7.12342', '7.12355', '7.12466', '7.12470', '7.12567', '7.12580', '7.12753', '7.12756', '7.12757', '7.12769', '7.13443', '7.13448', '7.13975', '7.14252', '7.14292', '7.14737', '7.14815', '7.14821', '7.15036', '7.15053', '7.15492', '7.16259', '7.16539', '7.16553', '7.16768', '7.16831', '7.16904', '7.17040', '7.17049', '7.17058', '7.17190', '7.17198', '7.17256', '7.17543', '7.17774', '7.18073', '7.18401', '7.20883', '7.20885', '7.20989', '7.21125', '7.21126', '7.21185', '7.21222', '7.21229', '7.21230', '7.21232', '7.21301', '7.21347', '7.21409', '7.21430', '7.21495', '7.21517', '7.21536', '7.21606', '7.21646', '7.21686', '7.21706', '7.22098', '7.22102', '7.22327', '7.22436', '7.22488', '7.22500', '7.22639', '7.22656', '7.22848', '7.22868', '7.22889', '7.22923', '7.23072', '7.23097', '7.23300', '7.23557', '7.23650', '7.23651', '7.24152', '7.24164', '7.24247', '7.24253', '7.24325', '7.24587', '7.24599', '7.24600', '7.24607', '7.24618', '7.24677', '7.24811', '7.24818', '7.24819', '7.24887', '7.24928', '7.25033', '7.25047', '7.25079', '7.25129', '7.25342', '7.25583', '7.25641', '7.26196', '7.26451', '7.26459', '7.26554', '7.26559', '7.26580', '7.26650', '7.26694', '7.26709', '7.26740', '7.26744', '7.26750', '7.26760', '7.26762', '7.26942', '7.26948', '7.26950', '7.27000', '7.27003', '7.27175', '7.27178', '7.27191', '7.27282', '7.27362', '7.27368', '7.27377', '7.27509', '7.27638', '7.27640', '7.27708', '7.27714', '7.27718', '7.27787', '7.27797', '7.27808', '7.27904', '7.28166', '7.28284', '7.28371', '7.28373', '7.28387', '7.28410', '7.28414', '7.28511', '7.28540', '7.28594', '7.28739', '7.28935', '7.28954', '7.28972', '7.29459', '7.29518', '7.29540', '7.29643', '7.29645', '7.29740', '7.30097', '7.30152', '7.30407', '7.30417', '7.30425', '7.30435', '7.30509', '7.30573', '7.30576', '7.30577', '7.30583', '7.30584', '7.30587', '7.30712', '7.30715', '7.30759', '7.30819', '7.30879', '7.30899', '7.30912', '7.30913', '7.30928', '7.30929', '7.30930', '7.30945', '7.30947', '7.30972', '7.30997', '7.31023', '7.31079', '7.31502', '7.31538', '7.31744', '7.31745', '7.31768', '7.31801', '7.31818', '7.31894', '7.32055', '7.32057', '7.32085', '7.32118', '7.32119', '7.32158', '7.32260', '7.32488', '7.32492', '7.32493', '7.32572', '7.32619', '7.32748', '7.32753', '7.32794', '7.32801', '7.32802', '7.32810', '7.32893', '7.32896', '7.33029', '7.33087', '7.33153', '7.33160', '7.33164', '7.33270', '7.33285', '7.33298', '7.33306', '7.33405', '7.33435', '7.33437', '7.33629', '7.33807', '7.33808', '7.33809', '7.33812', '7.33826', '7.33869', '7.33978', '7.34181', '7.34194', '7.34260', '7.34386', '7.34503', '7.34562', '7.34617', '7.34728', '7.34731', '7.34779', '7.34826', '7.34950', '7.34952', '7.34958', '7.34963', '7.34973', '7.34977', '7.34991', '7.35014', '7.35056', '7.35070', '7.35072', '7.35078', '7.35079', '7.35100', '7.35127', '7.35138', '7.35172', '7.35185', '7.35735', '7.35745', '7.35876', '7.35902', '7.35944', '7.35995', '7.36062', '7.36090', '7.36112', '7.36115', '7.36180', '7.36210', '7.36233', '7.36240', '7.36998', '7.37129', '7.37189', '7.37250', '7.37285', '7.37295', '7.37323', '7.37329', '7.37340', '7.37348', '7.37354', '7.37396', '7.37397', '7.37410', '7.37424', '7.37429', '7.37442', '7.37443', '7.37446', '7.37475', '7.37494', '7.37498', '7.37559', '7.37608', '7.37630', '7.37653', '7.37656', '7.37724', '7.37758', '7.37766', '7.37827', '7.37850', '7.37893', '7.37894', '7.37911', '7.37912', '7.37946', '7.37947', '7.38043', '7.38059', '7.38099', '7.38123', '7.38191', '7.38219', '7.38270', '7.38282', '7.38285', '7.38312', '7.38317', '7.38379', '7.38397', '7.38425', '7.38466', '7.38467', '7.38472', '7.38474', '7.38482', '7.38488', '7.38495', '7.38505', '7.38532', '7.38547', '7.38559', '7.38563', '7.38574', '7.38578', '7.38591', '7.38597', '7.38639', '7.38663', '7.38670', '7.38671', '7.38672', '7.38678', '7.38691', '7.38701', '7.38703', '7.38709', '7.38713', '7.38777', '7.38800', '7.38804', '7.38842', '7.38851', '7.38853', '7.38865', '7.38874', '7.38881', '7.38885', '7.38909', '7.38910', '7.38911', '7.38912', '7.38922', '7.38924', '7.38928', '7.38934', '7.38954', '7.39261', '7.39348', '7.39362', '7.39385', '7.39390', '7.39411', '7.39419', '7.39436', '7.39471', '7.39490', '7.39527', '7.39614', '7.39618', '7.39710', '7.39772', '7.39814', '7.39925', '7.40262', '7.40264', '7.40275', '7.40320', '7.40339', '7.40374', '7.40376', '7.40384', '7.40402', '7.40405', '7.40436', '7.40448', '7.40469', '7.40500', '7.40525', '7.40559', '7.40592', '7.40645', '7.40646', '7.40650', '7.40713', '7.40715', '7.40717', '7.40754', '7.40765', '7.40771', '7.40801', '7.40802', '7.40804', '7.40806', '7.41020', '7.41026', '7.41032', '7.41081', '7.41123', '7.41136', '7.41160', '7.41173', '7.41176', '7.41409', '7.41414', '7.41773', '7.41783', '7.41926', '7.41944', '7.42012', '7.42031', '7.42038', '7.42231', '7.42319', '7.42325', '7.42449', '7.42454', '7.42483', '7.42485', '7.42495', '7.42554', '7.42600', '7.42769', '7.42782', '7.42786', '7.42800', '7.42807', '7.42811', '7.42817', '7.42824', '7.42833', '7.42851', '7.42866', '7.42871', '7.42925', '7.42964', '7.42980', '7.42984', '7.42990', '7.43000', '7.43041', '7.43069', '7.43086', '7.43173', '7.43180', '7.43214', '7.43231', '7.43235', '7.43247', '7.43275', '7.43304', '7.43306', '7.43308', '7.43337', '7.43343', '7.43389', '7.43415', '7.43420', '7.43458', '7.43477', '7.43542', '7.43612', '7.43613', '7.43618', '7.43785', '7.43823', '7.43839', '7.43933', '7.43953', '7.44071', '7.44249', '7.44672', '7.44680', '7.44715', '7.44759', '7.44777', '7.44785', '7.44892', '7.44893', '7.44908', '7.44909', '7.44930', '7.44931']
Outer characteristic polynomial of the knot is: t^6+16t^4+18t^2+1
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.82', '7.13522', '7.32803']
2-strand cable arrow polynomial of the knot is: 384*K1**4*K2 - 1312*K1**4 + 128*K1**3*K2*K3 - 192*K1**3*K3 + 512*K1**2*K2**3 - 2192*K1**2*K2**2 - 256*K1**2*K2*K4 + 2312*K1**2*K2 - 160*K1**2*K3**2 - 544*K1**2 + 256*K1*K2**3*K3 - 320*K1*K2**2*K3 - 64*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 1760*K1*K2*K3 + 152*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 600*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 456*K2**2*K4 - 518*K2**2 + 136*K2*K3*K5 + 16*K2*K4*K6 - 292*K3**2 - 58*K4**2 - 12*K5**2 - 2*K6**2 + 720
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.82']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.2', 'vk5.7', 'vk5.12', 'vk5.46', 'vk5.51', 'vk5.56', 'vk5.89', 'vk5.113', 'vk5.124', 'vk5.133', 'vk5.591', 'vk5.596', 'vk5.1016', 'vk5.1022', 'vk5.1339', 'vk5.1344']
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 O1O2O3U1U3O4O5U4U5U2
R3 orbit {'O1O2O3U1U3O4O5U4U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4U5O4O5U1U3
Gauss code of K* O1O2O3U4U3U5O4O5U1U2
Gauss code of -K* O1O2O3U2U3O4O5U4U1U5
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 1 1 -1 1],[ 2 0 2 1 0 0],[-1 -2 0 0 -1 1],[-1 -1 0 0 0 0],[ 1 0 1 0 0 1],[-1 0 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 -1 -2],[-1 0 1 0 -1 -2],[-1 -1 0 0 -1 0],[-1 0 0 0 0 -1],[ 1 1 1 0 0 0],[ 2 2 0 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,1,2,-1,0,1,2,0,1,0,0,1,0]
Phi over symmetry [-2,-1,1,1,1,0,0,1,2,1,0,1,0,-1,0]
Phi of -K [-2,-1,1,1,1,1,1,2,3,1,2,1,0,-1,0]
Phi of K* [-1,-1,-1,1,2,-1,0,1,3,0,1,1,2,2,1]
Phi of -K* [-2,-1,1,1,1,0,0,1,2,1,0,1,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial -3z^2-14z-15
Enhanced Jones-Krushkal polynomial -3w^3z^2-14w^2z-15w
Inner characteristic polynomial t^5+8t^3+7t
Outer characteristic polynomial t^6+16t^4+18t^2+1
Flat arrow polynomial -4*K1**3 + 6*K1**2 + 4*K1*K2 + K1 - 3*K2 - K3 - 2
2-strand cable arrow polynomial 384*K1**4*K2 - 1312*K1**4 + 128*K1**3*K2*K3 - 192*K1**3*K3 + 512*K1**2*K2**3 - 2192*K1**2*K2**2 - 256*K1**2*K2*K4 + 2312*K1**2*K2 - 160*K1**2*K3**2 - 544*K1**2 + 256*K1*K2**3*K3 - 320*K1*K2**2*K3 - 64*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 1760*K1*K2*K3 + 152*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 600*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 456*K2**2*K4 - 518*K2**2 + 136*K2*K3*K5 + 16*K2*K4*K6 - 292*K3**2 - 58*K4**2 - 12*K5**2 - 2*K6**2 + 720
Genus of based matrix 1
Fillings of based matrix [[{1, 5}, {2, 4}, {3}], [{2, 5}, {4}, {1, 3}], [{3, 5}, {2, 4}, {1}], [{4, 5}, {1, 3}, {2}], [{4, 5}, {2, 3}, {1}], [{4, 5}, {3}, {1, 2}], [{5}, {2, 4}, {1, 3}]]
If K is slice False
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