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Flat knot 6.1551

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,0,1,0,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1551']
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+65t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1551']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 224*K1**4*K2 - 1152*K1**4 + 32*K1**3*K2*K3 - 832*K1**2*K2**2 + 1928*K1**2*K2 - 768*K1**2*K3**2 - 112*K1**2*K4**2 - 1052*K1**2 + 1488*K1*K2*K3 + 760*K1*K3*K4 + 152*K1*K4*K5 - 56*K2**4 - 64*K2**2*K3**2 - 16*K2**2*K4**2 + 112*K2**2*K4 - 1004*K2**2 + 120*K2*K3*K5 + 32*K2*K4*K6 - 568*K3**2 - 262*K4**2 - 92*K5**2 - 12*K6**2 + 1204
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1551']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4206', 'vk6.4287', 'vk6.5463', 'vk6.5576', 'vk6.7573', 'vk6.7665', 'vk6.9075', 'vk6.9156', 'vk6.11170', 'vk6.12256', 'vk6.12365', 'vk6.19383', 'vk6.19676', 'vk6.19774', 'vk6.26165', 'vk6.26213', 'vk6.26581', 'vk6.26656', 'vk6.30768', 'vk6.31289', 'vk6.31686', 'vk6.31973', 'vk6.32447', 'vk6.32864', 'vk6.38173', 'vk6.38197', 'vk6.39093', 'vk6.41347', 'vk6.44834', 'vk6.44942', 'vk6.45849', 'vk6.48516', 'vk6.49318', 'vk6.52305', 'vk6.53149', 'vk6.58447', 'vk6.62971', 'vk6.63584', 'vk6.66325', 'vk6.66353']
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 O1O2O3U4O5U2U6O4O6U5U1U3
R3 orbit {'O1O2O3U4U1O5U6O4O6U2U5U3', 'O1O2O3U4O5U2U6O4O6U5U1U3'}
R3 orbit length 2
Gauss code of -K O1O2O3U1U3U4O5O6U5U2O4U6
Gauss code of K* O1O2O3U2U4U3O5U1O4O6U5U6
Gauss code of -K* O1O2O3U4U5O4O6U3O5U1U6U2
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 2 -1 0 1],[ 1 0 0 2 0 0 2],[ 1 0 0 1 1 0 2],[-2 -2 -1 0 -2 -1 -1],[ 1 0 -1 2 0 1 1],[ 0 0 0 1 -1 0 0],[-1 -2 -2 1 -1 0 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 0 -2 -1 -2],[ 0 1 0 0 0 -1 0],[ 1 1 2 0 0 1 0],[ 1 2 1 1 -1 0 0],[ 1 2 2 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,1,1,1,2,2,0,2,1,2,0,1,0,-1,0,0]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,0,1,0,1,0,0,-1]
Phi of -K [-1,-1,-1,0,1,2,-1,0,1,0,2,0,0,1,1,1,0,1,1,1,0]
Phi of K* [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,0,1,0,1,0,0,-1]
Phi of -K* [-1,-1,-1,0,1,2,-1,0,1,1,2,0,0,2,1,0,2,2,0,1,1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 11z+23
Enhanced Jones-Krushkal polynomial 11w^2z+23w
Inner characteristic polynomial t^6+22t^4+44t^2
Outer characteristic polynomial t^7+30t^5+65t^3
Flat arrow polynomial -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4
2-strand cable arrow polynomial -64*K1**6 + 224*K1**4*K2 - 1152*K1**4 + 32*K1**3*K2*K3 - 832*K1**2*K2**2 + 1928*K1**2*K2 - 768*K1**2*K3**2 - 112*K1**2*K4**2 - 1052*K1**2 + 1488*K1*K2*K3 + 760*K1*K3*K4 + 152*K1*K4*K5 - 56*K2**4 - 64*K2**2*K3**2 - 16*K2**2*K4**2 + 112*K2**2*K4 - 1004*K2**2 + 120*K2*K3*K5 + 32*K2*K4*K6 - 568*K3**2 - 262*K4**2 - 92*K5**2 - 12*K6**2 + 1204
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{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}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]]
If K is slice False
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