Table of flat knot invariants
Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo
Glossary Reference List

Flat knot 5.42

Min(phi) over symmetries of the knot is: [-3,0,0,1,2,0,1,3,3,0,1,0,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['5.42', '7.28336']
Arrow polynomial of the knot is: 2*K1**2 + 2*K1*K2 - K1 - K2 - K3
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.18', '5.20', '5.39', '5.42', '5.51', '5.55', '5.59', '7.332', '7.353', '7.709', '7.861', '7.921', '7.2228', '7.2240', '7.2250', '7.2252', '7.2594', '7.2599', '7.2641', '7.2645', '7.2780', '7.2960', '7.2969', '7.3000', '7.3004', '7.3328', '7.3583', '7.3641', '7.4534', '7.4539', '7.4676', '7.4745', '7.4963', '7.5142', '7.5183', '7.5202', '7.5464', '7.5653', '7.6482', '7.6486', '7.6537', '7.6648', '7.6733', '7.6862', '7.8529', '7.9104', '7.9108', '7.9118', '7.9664', '7.9670', '7.9675', '7.9684', '7.9706', '7.9773', '7.9900', '7.9911', '7.9943', '7.9949', '7.10023', '7.10069', '7.10082', '7.10135', '7.10336', '7.10632', '7.10652', '7.10658', '7.11017', '7.11029', '7.11031', '7.11312', '7.11471', '7.11698', '7.11704', '7.11709', '7.11796', '7.11821', '7.11851', '7.11858', '7.11860', '7.11867', '7.11876', '7.11877', '7.11879', '7.11883', '7.11884', '7.11887', '7.11927', '7.11980', '7.12008', '7.12037', '7.12057', '7.12125', '7.12160', '7.12329', '7.12336', '7.12349', '7.12364', '7.12387', '7.12389', '7.12465', '7.12555', '7.12562', '7.12741', '7.12754', '7.12816', '7.12851', '7.12852', '7.12864', '7.12865', '7.12968', '7.13008', '7.13429', '7.13456', '7.13555', '7.13557', '7.13602', '7.13771', '7.13819', '7.13822', '7.13850', '7.13911', '7.13936', '7.13942', '7.14083', '7.14120', '7.14158', '7.14188', '7.14211', '7.14421', '7.14830', '7.14876', '7.14967', '7.15005', '7.15106', '7.15114', '7.15118', '7.15134', '7.15163', '7.15340', '7.15355', '7.16163', '7.16198', '7.16227', '7.16397', '7.16449', '7.16467', '7.16474', '7.16990', '7.16996', '7.17104', '7.17108', '7.17130', '7.17298', '7.17302', '7.17303', '7.17324', '7.17493', '7.17495', '7.17521', '7.17618', '7.17700', '7.18072', '7.18268', '7.18526', '7.18573', '7.18936', '7.19383', '7.19392', '7.19394', '7.19468', '7.19479', '7.19488', '7.19558', '7.19576', '7.19869', '7.20096', '7.20176', '7.20196', '7.20220', '7.20233', '7.20513', '7.20519', '7.20520', '7.20522', '7.20554', '7.20555', '7.20568', '7.20576', '7.20607', '7.20866', '7.20888', '7.20913', '7.20974', '7.21005', '7.21027', '7.21028', '7.21029', '7.21030', '7.21068', '7.21072', '7.21091', '7.21093', '7.21343', '7.21344', '7.21345', '7.21394', '7.21400', '7.21421', '7.21422', '7.21423', '7.21469', '7.21473', '7.21730', '7.21761', '7.21768', '7.21798', '7.21811', '7.21826', '7.21830', '7.21837', '7.21854', '7.21880', '7.21911', '7.21922', '7.21942', '7.22024', '7.22031', '7.22259', '7.22496', '7.22599', '7.22624', '7.22705', '7.22863', '7.22883', '7.22973', '7.22978', '7.22992', '7.23292', '7.23381', '7.23550', '7.23852', '7.23937', '7.23967', '7.24013', '7.24014', '7.24093', '7.24336', '7.24337', '7.24339', '7.24340', '7.24345', '7.24355', '7.24359', '7.24485', '7.24639', '7.24866', '7.24897', '7.24943', '7.24980', '7.25133', '7.25135', '7.25233', '7.25276', '7.25284', '7.25520', '7.25521', '7.25522', '7.25538', '7.25544', '7.25789', '7.26064', '7.26076', '7.26297', '7.26320', '7.26378', '7.26502', '7.26525', '7.26563', '7.26589', '7.26813', '7.26824', '7.26869', '7.27227', '7.27433', '7.27492', '7.27496', '7.27703', '7.27773', '7.27818', '7.27994', '7.28019', '7.28039', '7.28086', '7.28099', '7.28116', '7.28640', '7.28667', '7.28821', '7.28903', '7.28942', '7.29025', '7.29080', '7.29084', '7.29106', '7.29138', '7.29242', '7.29292', '7.29304', '7.29369', '7.29442', '7.29445', '7.29467', '7.29676', '7.29725', '7.29786', '7.29851', '7.29990', '7.30050', '7.30231', '7.30252', '7.30278', '7.30279', '7.30358', '7.30813', '7.30943', '7.31002', '7.31065', '7.31214', '7.31534', '7.31570', '7.31586', '7.31628', '7.31630', '7.31717', '7.31728', '7.31858', '7.31864', '7.31879', '7.31984', '7.32065', '7.32989', '7.33207', '7.33415', '7.33467', '7.33473', '7.34215', '7.34224', '7.34431']
Outer characteristic polynomial of the knot is: t^6+37t^4+25t^2+1
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.42']
2-strand cable arrow polynomial of the knot is: -64*K1**4 + 32*K1**3*K2*K3 - 368*K1**2*K2**2 - 160*K1**2*K2*K4 + 832*K1**2*K2 - 64*K1**2*K3**2 - 872*K1**2 - 160*K1*K2**2*K3 + 880*K1*K2*K3 + 328*K1*K3*K4 - 8*K2**4 - 48*K2**2*K3**2 - 72*K2**2*K4**2 + 352*K2**2*K4 - 798*K2**2 + 32*K2*K3*K5 + 40*K2*K4*K6 - 364*K3**2 - 230*K4**2 - 4*K5**2 - 2*K6**2 + 716
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.42']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.447', 'vk5.475', 'vk5.648', 'vk5.794', 'vk5.930', 'vk5.975', 'vk5.1163', 'vk5.1308', 'vk5.1506', 'vk5.1542', 'vk5.1655', 'vk5.1778', 'vk5.1800', 'vk5.1820', 'vk5.1878', 'vk5.1939']
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 O1O2O3O4U1U4O5U3U2U5
R3 orbit {'O1O2O3O4U1U4O5U3U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3U2O5U1U4
Gauss code of K* O1O2O3U4U2U1U5O4O5U3
Gauss code of -K* O1O2O3U1O4O5U4U3U2U5
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 -3 0 0 1 2],[ 3 0 3 2 1 2],[ 0 -3 0 0 0 2],[ 0 -2 0 0 0 1],[-1 -1 0 0 0 0],[-2 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -3],[-2 0 0 -1 -2 -2],[-1 0 0 0 0 -1],[ 0 1 0 0 0 -2],[ 0 2 0 0 0 -3],[ 3 2 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,3,0,1,2,2,0,0,1,0,2,3]
Phi over symmetry [-3,0,0,1,2,0,1,3,3,0,1,0,1,1,1]
Phi of -K [-3,0,0,1,2,0,1,3,3,0,1,0,1,1,1]
Phi of K* [-2,-1,0,0,3,1,0,1,3,1,1,3,0,0,1]
Phi of -K* [-3,0,0,1,2,2,3,1,2,0,0,1,0,2,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial -3z^2-14z-15
Enhanced Jones-Krushkal polynomial -3w^3z^2-14w^2z-15w
Inner characteristic polynomial t^5+23t^3+6t
Outer characteristic polynomial t^6+37t^4+25t^2+1
Flat arrow polynomial 2*K1**2 + 2*K1*K2 - K1 - K2 - K3
2-strand cable arrow polynomial -64*K1**4 + 32*K1**3*K2*K3 - 368*K1**2*K2**2 - 160*K1**2*K2*K4 + 832*K1**2*K2 - 64*K1**2*K3**2 - 872*K1**2 - 160*K1*K2**2*K3 + 880*K1*K2*K3 + 328*K1*K3*K4 - 8*K2**4 - 48*K2**2*K3**2 - 72*K2**2*K4**2 + 352*K2**2*K4 - 798*K2**2 + 32*K2*K3*K5 + 40*K2*K4*K6 - 364*K3**2 - 230*K4**2 - 4*K5**2 - 2*K6**2 + 716
Genus of based matrix 2
Fillings of based matrix [[{1, 5}, {2, 4}, {3}], [{1, 5}, {3, 4}, {2}], [{1, 5}, {4}, {2, 3}], [{1, 5}, {4}, {3}, {2}], [{2, 5}, {1, 4}, {3}], [{2, 5}, {3, 4}, {1}], [{2, 5}, {4}, {1, 3}], [{2, 5}, {4}, {3}, {1}], [{3, 5}, {1, 4}, {2}], [{3, 5}, {2, 4}, {1}], [{3, 5}, {4}, {1, 2}], [{3, 5}, {4}, {2}, {1}], [{4, 5}, {1, 3}, {2}], [{4, 5}, {2, 3}, {1}], [{4, 5}, {3}, {1, 2}], [{4, 5}, {3}, {2}, {1}], [{5}, {1, 4}, {2, 3}], [{5}, {1, 4}, {3}, {2}], [{5}, {2, 4}, {1, 3}], [{5}, {2, 4}, {3}, {1}], [{5}, {3, 4}, {1, 2}], [{5}, {3, 4}, {2}, {1}], [{5}, {4}, {1, 3}, {2}], [{5}, {4}, {2, 3}, {1}], [{5}, {4}, {3}, {1, 2}]]
If K is slice False
Contact