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Flat knot 6.1678

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,1,1,1,1,1,1,2,2,0,-1,-1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1678']
Arrow polynomial of the knot is: -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.546', '6.591', '6.598', '6.666', '6.680', '6.742', '6.778', '6.805', '6.822', '6.824', '6.1129', '6.1512', '6.1647', '6.1678', '6.1705', '6.1847', '6.1857']
Outer characteristic polynomial of the knot is: t^7+28t^5+30t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1678']
2-strand cable arrow polynomial of the knot is: -448*K1**6 - 256*K1**4*K2**2 + 768*K1**4*K2 - 2816*K1**4 + 224*K1**3*K2*K3 - 3424*K1**2*K2**2 + 4648*K1**2*K2 - 1120*K1**2*K3**2 - 112*K1**2*K4**2 - 1028*K1**2 + 3616*K1*K2*K3 + 896*K1*K3*K4 + 88*K1*K4*K5 - 896*K2**4 - 480*K2**2*K3**2 - 48*K2**2*K4**2 + 624*K2**2*K4 - 1260*K2**2 + 312*K2*K3*K5 + 32*K2*K4*K6 - 816*K3**2 - 248*K4**2 - 52*K5**2 - 4*K6**2 + 1798
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1678']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11475', 'vk6.11779', 'vk6.12794', 'vk6.13130', 'vk6.17036', 'vk6.17279', 'vk6.20866', 'vk6.20943', 'vk6.22274', 'vk6.22355', 'vk6.23761', 'vk6.28340', 'vk6.31239', 'vk6.31590', 'vk6.32810', 'vk6.35539', 'vk6.35990', 'vk6.39968', 'vk6.40111', 'vk6.42040', 'vk6.42948', 'vk6.43243', 'vk6.46508', 'vk6.46631', 'vk6.52240', 'vk6.53074', 'vk6.53394', 'vk6.55457', 'vk6.58856', 'vk6.59938', 'vk6.64415', 'vk6.69730']
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 O1O2O3U2O4U5U1U3O6O5U6U4
R3 orbit {'O1O2O3U2O4U5U1U3O6O5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O6O5U1U3U6O4U2
Gauss code of K* O1O2O3U4U1O4O5U2U6U3O6U5
Gauss code of -K* O1O2O3U4O5U1U5U2O4O6U3U6
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 2 -1 -1],[ 1 0 0 2 1 0 -1],[ 1 0 0 1 1 0 0],[-2 -2 -1 0 0 -2 -1],[-2 -1 -1 0 0 -1 -1],[ 1 0 0 2 1 0 -1],[ 1 1 0 1 1 1 0]]
Primitive based matrix [[ 0 2 2 -1 -1 -1 -1],[-2 0 0 -1 -1 -1 -1],[-2 0 0 -1 -1 -2 -2],[ 1 1 1 0 0 1 1],[ 1 1 1 0 0 0 0],[ 1 1 2 -1 0 0 0],[ 1 1 2 -1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,1,1,1,1,0,1,1,1,1,1,1,2,2,0,-1,-1,0,0,0]
Phi over symmetry [-2,-2,1,1,1,1,0,1,1,1,1,1,1,2,2,0,-1,-1,0,0,0]
Phi of -K [-1,-1,-1,-1,2,2,-1,-1,0,2,2,0,0,1,2,0,1,2,2,2,0]
Phi of K* [-2,-2,1,1,1,1,0,1,1,2,2,2,2,2,2,0,-1,0,-1,0,0]
Phi of -K* [-1,-1,-1,-1,2,2,-1,0,0,1,2,0,1,1,1,0,1,1,1,2,0]
Symmetry type of based matrix c
u-polynomial -2t^2+4t
Normalized Jones-Krushkal polynomial 14z+29
Enhanced Jones-Krushkal polynomial 14w^2z+29w
Inner characteristic polynomial t^6+16t^4+8t^2
Outer characteristic polynomial t^7+28t^5+30t^3
Flat arrow polynomial -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -448*K1**6 - 256*K1**4*K2**2 + 768*K1**4*K2 - 2816*K1**4 + 224*K1**3*K2*K3 - 3424*K1**2*K2**2 + 4648*K1**2*K2 - 1120*K1**2*K3**2 - 112*K1**2*K4**2 - 1028*K1**2 + 3616*K1*K2*K3 + 896*K1*K3*K4 + 88*K1*K4*K5 - 896*K2**4 - 480*K2**2*K3**2 - 48*K2**2*K4**2 + 624*K2**2*K4 - 1260*K2**2 + 312*K2*K3*K5 + 32*K2*K4*K6 - 816*K3**2 - 248*K4**2 - 52*K5**2 - 4*K6**2 + 1798
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{3, 6}, {5}, {4}, {2}, {1}]]
If K is slice False
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