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Flat knot 6.1581

Min(phi) over symmetries of the knot is: [-2,0,1,1,1,1,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1581']
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^5+11t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1581']
2-strand cable arrow polynomial of the knot is: -576*K1**6 + 1408*K1**4*K2 - 3776*K1**4 + 288*K1**3*K2*K3 - 448*K1**3*K3 - 3136*K1**2*K2**2 - 224*K1**2*K2*K4 + 7184*K1**2*K2 - 1440*K1**2*K3**2 - 64*K1**2*K3*K5 - 432*K1**2*K4**2 - 4356*K1**2 - 480*K1*K2**2*K3 - 32*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 5776*K1*K2*K3 + 2400*K1*K3*K4 + 664*K1*K4*K5 - 152*K2**4 - 192*K2**2*K3**2 - 112*K2**2*K4**2 + 848*K2**2*K4 - 4188*K2**2 + 488*K2*K3*K5 + 136*K2*K4*K6 + 24*K3**2*K6 - 2440*K3**2 - 1094*K4**2 - 316*K5**2 - 52*K6**2 + 4612
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1581']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4130', 'vk6.4161', 'vk6.5368', 'vk6.5399', 'vk6.7498', 'vk6.7525', 'vk6.8999', 'vk6.9030', 'vk6.12422', 'vk6.12455', 'vk6.13352', 'vk6.13573', 'vk6.13604', 'vk6.14263', 'vk6.14710', 'vk6.14750', 'vk6.15196', 'vk6.15866', 'vk6.15906', 'vk6.30827', 'vk6.30860', 'vk6.32011', 'vk6.32044', 'vk6.33078', 'vk6.33109', 'vk6.33846', 'vk6.34308', 'vk6.48476', 'vk6.50261', 'vk6.53540', 'vk6.53929', 'vk6.54267']
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 O1O2O3U4O5U2U5O6O4U6U1U3
R3 orbit {'O1O2O3U4O5U2U5O6O4U6U1U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U3U4O5O4U6U2O6U5
Gauss code of K* O1O2O3U2U4U3O5U6O4O6U1U5
Gauss code of -K* O1O2O3U4U3O5O6U5O4U1U6U2
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 -1 -1 2 0 1 -1],[ 1 0 0 2 1 1 -1],[ 1 0 0 1 0 1 -1],[-2 -2 -1 0 -1 0 -1],[ 0 -1 0 1 0 1 -1],[-1 -1 -1 0 -1 0 0],[ 1 1 1 1 1 0 0]]
Primitive based matrix [[ 0 2 0 -1 -1],[-2 0 -1 -1 -1],[ 0 1 0 0 -1],[ 1 1 0 0 -1],[ 1 1 1 1 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,0,1,1,1,1,1,0,1,1]
Phi over symmetry [-2,0,1,1,1,1,1,0,1,1]
Phi of -K [-1,-1,0,2,-1,0,2,1,2,1]
Phi of K* [-2,0,1,1,1,2,2,0,1,1]
Phi of -K* [-1,-1,0,2,-1,0,1,1,1,1]
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^4+5t^2
Outer characteristic polynomial t^5+11t^3+7t
Flat arrow polynomial -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4
2-strand cable arrow polynomial -576*K1**6 + 1408*K1**4*K2 - 3776*K1**4 + 288*K1**3*K2*K3 - 448*K1**3*K3 - 3136*K1**2*K2**2 - 224*K1**2*K2*K4 + 7184*K1**2*K2 - 1440*K1**2*K3**2 - 64*K1**2*K3*K5 - 432*K1**2*K4**2 - 4356*K1**2 - 480*K1*K2**2*K3 - 32*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 5776*K1*K2*K3 + 2400*K1*K3*K4 + 664*K1*K4*K5 - 152*K2**4 - 192*K2**2*K3**2 - 112*K2**2*K4**2 + 848*K2**2*K4 - 4188*K2**2 + 488*K2*K3*K5 + 136*K2*K4*K6 + 24*K3**2*K6 - 2440*K3**2 - 1094*K4**2 - 316*K5**2 - 52*K6**2 + 4612
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{6}, {1, 5}, {4}, {2, 3}]]
If K is slice False
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