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Flat knot 5.102

Min(phi) over symmetries of the knot is: [-1,-1,0,1,1,-1,-1,1,2,1,0,0,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['5.102']
Arrow polynomial of the knot is: 4*K1**2 - 2*K2 - 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.24', '5.34', '5.47', '5.56', '5.62', '5.66', '5.67', '5.75', '5.89', '5.92', '5.94', '5.97', '5.102', '5.103', '5.105', '5.106', '5.110', '5.111', '5.115', '5.116', '5.117', '5.118', '5.119', '7.956', '7.5918', '7.8159', '7.9694', '7.10242', '7.10377', '7.10426', '7.10662', '7.11274', '7.11712', '7.12250', '7.12276', '7.13486', '7.14878', '7.15090', '7.16067', '7.16079', '7.16119', '7.16509', '7.16510', '7.17050', '7.17134', '7.17538', '7.17747', '7.18421', '7.18428', '7.18432', '7.18464', '7.18468', '7.18484', '7.18494', '7.18533', '7.18575', '7.18585', '7.18806', '7.18972', '7.18994', '7.18996', '7.19006', '7.19028', '7.19089', '7.19116', '7.19688', '7.19953', '7.20100', '7.20218', '7.20318', '7.20429', '7.20452', '7.21133', '7.21134', '7.21144', '7.21237', '7.21238', '7.21353', '7.21357', '7.21361', '7.21373', '7.21442', '7.21445', '7.21451', '7.21458', '7.21570', '7.21727', '7.21792', '7.22130', '7.22840', '7.22880', '7.23312', '7.23724', '7.23849', '7.24143', '7.24174', '7.24191', '7.24286', '7.24309', '7.24344', '7.24643', '7.24695', '7.24702', '7.24724', '7.24735', '7.24781', '7.24863', '7.24907', '7.24915', '7.24949', '7.24991', '7.25096', '7.25465', '7.25842', '7.25843', '7.25847', '7.25872', '7.25889', '7.25895', '7.25953', '7.25968', '7.25992', '7.25996', '7.25997', '7.26004', '7.26019', '7.26078', '7.26107', '7.26440', '7.26662', '7.26797', '7.26801', '7.26816', '7.27009', '7.27022', '7.27028', '7.27091', '7.27207', '7.27299', '7.27303', '7.27506', '7.27510', '7.27512', '7.27540', '7.27543', '7.27547', '7.27548', '7.27552', '7.27578', '7.27583', '7.27601', '7.27615', '7.27675', '7.27690', '7.27698', '7.27742', '7.27748', '7.27761', '7.27764', '7.27768', '7.27813', '7.27827', '7.27845', '7.27849', '7.27853', '7.27867', '7.27869', '7.27879', '7.27888', '7.27930', '7.28175', '7.28176', '7.28193', '7.28201', '7.28212', '7.28378', '7.28390', '7.28397', '7.28490', '7.28506', '7.28625', '7.28684', '7.28711', '7.28716', '7.28727', '7.28890', '7.28983', '7.29002', '7.29015', '7.29056', '7.29060', '7.29071', '7.29114', '7.29461', '7.29483', '7.29681', '7.29722', '7.30162', '7.30175', '7.30406', '7.30503', '7.30679', '7.30969', '7.31035', '7.31092', '7.31133', '7.31259', '7.31309', '7.31344', '7.31374', '7.31427', '7.31515', '7.31564', '7.31582', '7.31631', '7.31668', '7.31670', '7.31676', '7.31702', '7.31723', '7.31726', '7.31754', '7.31780', '7.31837', '7.32049', '7.32075', '7.32076', '7.32153', '7.32208', '7.32222', '7.32508', '7.32512', '7.32513', '7.32518', '7.32543', '7.32555', '7.32584', '7.32585', '7.32591', '7.32601', '7.32606', '7.32633', '7.32684', '7.32687', '7.32704', '7.32732', '7.32761', '7.32770', '7.32775', '7.32791', '7.32825', '7.32827', '7.32832', '7.32851', '7.32890', '7.32897', '7.32902', '7.32913', '7.32941', '7.32970', '7.32996', '7.32998', '7.33042', '7.33045', '7.33070', '7.33081', '7.33110', '7.33118', '7.33146', '7.33174', '7.33187', '7.33197', 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Outer characteristic polynomial of the knot is: t^6+14t^4+16t^2
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.102']
2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 320*K1**4 - 352*K1**2*K2**2 + 952*K1**2*K2 - 680*K1**2 + 456*K1*K2*K3 - 48*K2**4 + 32*K2**2*K4 - 480*K2**2 - 152*K3**2 - 4*K4**2 + 498
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.102']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.169', 'vk5.210', 'vk5.304', 'vk5.358', 'vk5.484', 'vk5.705', 'vk5.712', 'vk5.859', 'vk5.866', 'vk5.945', 'vk5.990', 'vk5.1220', 'vk5.1386', 'vk5.1445', 'vk5.1550', 'vk5.1898']
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 O1O2O3U2U1O4U3O5U4U5
R3 orbit {'O1O2O3U2U1O4U3O5U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U1O5U3U2
Gauss code of K* O1O2U3O4U5O3O5U2U1U4
Gauss code of -K* O1O2U1O3U2O4O5U3U5U4
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 -1 -1 1 0 1],[ 1 0 0 2 1 0],[ 1 0 0 1 1 0],[-1 -2 -1 0 1 1],[ 0 -1 -1 -1 0 1],[-1 0 0 -1 -1 0]]
Primitive based matrix [[ 0 1 1 0 -1 -1],[-1 0 1 1 -1 -2],[-1 -1 0 -1 0 0],[ 0 -1 1 0 -1 -1],[ 1 1 0 1 0 0],[ 1 2 0 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,1,1,-1,-1,1,2,1,0,0,1,1,0]
Phi over symmetry [-1,-1,0,1,1,-1,-1,1,2,1,0,0,1,1,0]
Phi of -K [-1,-1,0,1,1,0,0,0,2,0,1,2,2,0,-1]
Phi of K* [-1,-1,0,1,1,-1,0,2,2,2,0,1,0,0,0]
Phi of -K* [-1,-1,0,1,1,0,1,0,1,1,0,2,1,-1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial -7z-13
Enhanced Jones-Krushkal polynomial 4w^3z-11w^2z-13w
Inner characteristic polynomial t^5+10t^3+2t
Outer characteristic polynomial t^6+14t^4+16t^2
Flat arrow polynomial 4*K1**2 - 2*K2 - 1
2-strand cable arrow polynomial 96*K1**4*K2 - 320*K1**4 - 352*K1**2*K2**2 + 952*K1**2*K2 - 680*K1**2 + 456*K1*K2*K3 - 48*K2**4 + 32*K2**2*K4 - 480*K2**2 - 152*K3**2 - 4*K4**2 + 498
Genus of based matrix 1
Fillings of based matrix [[{1, 5}, {4}, {2, 3}], [{2, 5}, {1, 4}, {3}], [{2, 5}, {3, 4}, {1}], [{2, 5}, {4}, {1, 3}]]
If K is slice False
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