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Flat knot 6.214

Min(phi) over symmetries of the knot is: [-3,-2,2,3,-1,2,4,1,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.206', '6.214', '7.15858', '7.31209']
Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932']
Outer characteristic polynomial of the knot is: t^5+65t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.206', '6.214']
2-strand cable arrow polynomial of the knot is: -544*K1**2*K2**2 + 336*K1**2*K2 - 32*K1**2*K3**2 - 648*K1**2 + 192*K1*K2**3*K3 + 1216*K1*K2*K3 + 112*K1*K3*K4 + 80*K1*K4*K5 - 144*K2**4 - 256*K2**2*K3**2 - 16*K2**2*K4**2 + 64*K2**2*K4 - 460*K2**2 + 160*K2*K3*K5 + 16*K2*K4*K6 - 32*K3**4 + 32*K3**2*K6 - 536*K3**2 - 108*K4**2 - 80*K5**2 - 12*K6**2 + 658
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.214']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71472', 'vk6.71488', 'vk6.71532', 'vk6.71547', 'vk6.72005', 'vk6.72058', 'vk6.72519', 'vk6.72627', 'vk6.72657', 'vk6.72778', 'vk6.72911', 'vk6.72950', 'vk6.73131', 'vk6.75766', 'vk6.75802', 'vk6.77096', 'vk6.77161', 'vk6.77454', 'vk6.77875', 'vk6.77929', 'vk6.78790', 'vk6.80382', 'vk6.81300', 'vk6.81422', 'vk6.86891', 'vk6.86899', 'vk6.87251', 'vk6.87738', 'vk6.87806', 'vk6.89148', 'vk6.89351', 'vk6.89507']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4O5U2O6U5U4U1U6U3
R3 orbit {'O1O2O3O4O5U2O6U5U4U1U6U3', 'O1O2O3O4U1O5U4U6U2U5O6U3', 'O1O2O3O4O5U2U4U6U5U1O6U3'}
R3 orbit length 3
Gauss code of -K O1O2O3O4O5U3U6U5U2U1O6U4
Gauss code of K* O1O2O3O4O5U3U6U5U2U1O6U4
Gauss code of -K* Same
Diagrammatic symmetry type -
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 -3 2 0 0 3],[ 2 0 -2 3 1 1 3],[ 3 2 0 3 2 1 2],[-2 -3 -3 0 -1 -1 2],[ 0 -1 -2 1 0 0 2],[ 0 -1 -1 1 0 0 1],[-3 -3 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 3 2 -2 -3],[-3 0 -2 -3 -2],[-2 2 0 -3 -3],[ 2 3 3 0 -2],[ 3 2 3 2 0]]
If based matrix primitive False
Phi of primitive based matrix [-3,-2,2,3,2,3,2,3,3,2]
Phi over symmetry [-3,-2,2,3,-1,2,4,1,2,-1]
Phi of -K [-3,-2,2,3,-1,2,4,1,2,-1]
Phi of K* [-3,-2,2,3,-1,2,4,1,2,-1]
Phi of -K* [-3,-2,2,3,2,3,2,3,3,2]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z+9
Enhanced Jones-Krushkal polynomial -6w^3z+10w^2z+9w
Inner characteristic polynomial t^4+39t^2+1
Outer characteristic polynomial t^5+65t^3+5t
Flat arrow polynomial -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -544*K1**2*K2**2 + 336*K1**2*K2 - 32*K1**2*K3**2 - 648*K1**2 + 192*K1*K2**3*K3 + 1216*K1*K2*K3 + 112*K1*K3*K4 + 80*K1*K4*K5 - 144*K2**4 - 256*K2**2*K3**2 - 16*K2**2*K4**2 + 64*K2**2*K4 - 460*K2**2 + 160*K2*K3*K5 + 16*K2*K4*K6 - 32*K3**4 + 32*K3**2*K6 - 536*K3**2 - 108*K4**2 - 80*K5**2 - 12*K6**2 + 658
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}]]
If K is slice True
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