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Flat knot 6.1711

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,1,0,0,1,1,1,0,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1711']
Arrow polynomial of the knot is: -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.239', '6.428', '6.470', '6.556', '6.700', '6.910', '6.962', '6.1006', '6.1013', '6.1038', '6.1207', '6.1224', '6.1225', '6.1269', '6.1270', '6.1308', '6.1319', '6.1320', '6.1323', '6.1485', '6.1551', '6.1579', '6.1581', '6.1660', '6.1672', '6.1679', '6.1711', '6.1719', '6.1732', '6.1745', '6.1748', '6.1827', '6.1836', '6.1838', '6.1850', '6.1866']
Outer characteristic polynomial of the knot is: t^7+30t^5+43t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1711']
2-strand cable arrow polynomial of the knot is: -384*K1**6 + 1248*K1**4*K2 - 5152*K1**4 + 384*K1**3*K2*K3 + 64*K1**3*K3*K4 - 928*K1**3*K3 - 2576*K1**2*K2**2 - 736*K1**2*K2*K4 + 8224*K1**2*K2 - 1568*K1**2*K3**2 - 160*K1**2*K3*K5 - 736*K1**2*K4**2 - 4640*K1**2 - 224*K1*K2**2*K3 - 32*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6008*K1*K2*K3 + 3280*K1*K3*K4 + 896*K1*K4*K5 - 24*K2**4 - 48*K2**2*K3**2 - 112*K2**2*K4**2 + 720*K2**2*K4 - 4276*K2**2 + 256*K2*K3*K5 + 136*K2*K4*K6 + 24*K3**2*K6 - 2692*K3**2 - 1382*K4**2 - 284*K5**2 - 52*K6**2 + 4988
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1711']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4465', 'vk6.4562', 'vk6.5847', 'vk6.5976', 'vk6.7905', 'vk6.8025', 'vk6.9334', 'vk6.9455', 'vk6.13415', 'vk6.13510', 'vk6.13703', 'vk6.14073', 'vk6.15044', 'vk6.15166', 'vk6.17780', 'vk6.17813', 'vk6.18822', 'vk6.19422', 'vk6.19717', 'vk6.24327', 'vk6.25415', 'vk6.25448', 'vk6.26596', 'vk6.33269', 'vk6.33328', 'vk6.37549', 'vk6.44879', 'vk6.48654', 'vk6.50552', 'vk6.53661', 'vk6.55804', 'vk6.65474']
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 O1O2O3U1U4O5O4U6U3O6U5U2
R3 orbit {'O1O2O3U1U4O5O4U6U3O6U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4O5U1U5O6O4U6U3
Gauss code of K* O1O2U1O3O4U5U4U2O5O6U3U6
Gauss code of -K* O1O2U3O4O3U5U2O5O6U4U1U6
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 0 -1],[ 2 0 2 1 2 1 1],[-1 -2 0 0 0 0 -2],[-1 -1 0 0 -1 -1 -1],[-1 -2 0 1 0 0 -2],[ 0 -1 0 1 0 0 0],[ 1 -1 2 1 2 0 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -2 -2],[-1 -1 0 0 -1 -1 -1],[-1 0 0 0 0 -2 -2],[ 0 0 1 0 0 0 -1],[ 1 2 1 2 0 0 -1],[ 2 2 1 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,0,2,2,0,1,1,1,0,2,2,0,1,1]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,1,2,1,0,0,1,1,1,0,0,-1,0]
Phi of -K [-2,-1,0,1,1,1,0,1,1,1,2,1,0,0,1,1,1,0,0,-1,0]
Phi of K* [-1,-1,-1,0,1,2,-1,0,0,1,2,0,1,0,1,1,0,1,1,1,0]
Phi of -K* [-2,-1,0,1,1,1,1,1,1,2,2,0,1,2,2,1,0,0,-1,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+22t^4+30t^2+4
Outer characteristic polynomial t^7+30t^5+43t^3+7t
Flat arrow polynomial -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4
2-strand cable arrow polynomial -384*K1**6 + 1248*K1**4*K2 - 5152*K1**4 + 384*K1**3*K2*K3 + 64*K1**3*K3*K4 - 928*K1**3*K3 - 2576*K1**2*K2**2 - 736*K1**2*K2*K4 + 8224*K1**2*K2 - 1568*K1**2*K3**2 - 160*K1**2*K3*K5 - 736*K1**2*K4**2 - 4640*K1**2 - 224*K1*K2**2*K3 - 32*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6008*K1*K2*K3 + 3280*K1*K3*K4 + 896*K1*K4*K5 - 24*K2**4 - 48*K2**2*K3**2 - 112*K2**2*K4**2 + 720*K2**2*K4 - 4276*K2**2 + 256*K2*K3*K5 + 136*K2*K4*K6 + 24*K3**2*K6 - 2692*K3**2 - 1382*K4**2 - 284*K5**2 - 52*K6**2 + 4988
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}]]
If K is slice False
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