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Flat knot 6.1285

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,1,1,3,0,1,0,1,0,1,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1285']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923']
Outer characteristic polynomial of the knot is: t^7+33t^5+26t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1285']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 1472*K1**4*K2**2 + 3648*K1**4*K2 - 6080*K1**4 + 1536*K1**3*K2*K3 - 1344*K1**3*K3 - 448*K1**2*K2**4 + 2880*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 10688*K1**2*K2**2 - 1216*K1**2*K2*K4 + 11552*K1**2*K2 - 992*K1**2*K3**2 - 112*K1**2*K4**2 - 3972*K1**2 + 1024*K1*K2**3*K3 - 1600*K1*K2**2*K3 - 384*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 8416*K1*K2*K3 + 1280*K1*K3*K4 + 192*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1872*K2**4 - 64*K2**3*K6 - 656*K2**2*K3**2 - 128*K2**2*K4**2 + 1728*K2**2*K4 - 3564*K2**2 + 560*K2*K3*K5 + 80*K2*K4*K6 - 1736*K3**2 - 560*K4**2 - 132*K5**2 - 12*K6**2 + 4262
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1285']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4701', 'vk6.5006', 'vk6.6195', 'vk6.6668', 'vk6.8188', 'vk6.8608', 'vk6.9566', 'vk6.9907', 'vk6.17393', 'vk6.20910', 'vk6.20978', 'vk6.22320', 'vk6.22400', 'vk6.23558', 'vk6.23897', 'vk6.28390', 'vk6.36161', 'vk6.40040', 'vk6.40171', 'vk6.42091', 'vk6.43070', 'vk6.43376', 'vk6.46572', 'vk6.46680', 'vk6.48733', 'vk6.49529', 'vk6.49734', 'vk6.51431', 'vk6.55559', 'vk6.58908', 'vk6.65297', 'vk6.69766']
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 O1O2O3U1O4O5U6U3O6U4U5U2
R3 orbit {'O1O2O3U1O4O5U6U3O6U4U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4U5O6U1U6O4O5U3
Gauss code of K* O1O2O3U4U3U5O4U1U2O6O5U6
Gauss code of -K* O1O2O3U4O5O4U2U3O6U5U1U6
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 0 0 2 -1],[ 2 0 2 1 1 1 1],[-1 -2 0 -1 0 2 -2],[ 0 -1 1 0 0 1 0],[ 0 -1 0 0 0 1 0],[-2 -1 -2 -1 -1 0 -2],[ 1 -1 2 0 0 2 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -2 -1 -1 -2 -1],[-1 2 0 0 -1 -2 -2],[ 0 1 0 0 0 0 -1],[ 0 1 1 0 0 0 -1],[ 1 2 2 0 0 0 -1],[ 2 1 2 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,2,1,1,2,1,0,1,2,2,0,0,1,0,1,1]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,1,1,3,0,1,0,1,0,1,1,1,1,0]
Phi of -K [-2,-1,0,0,1,2,0,1,1,1,3,1,1,0,1,0,0,1,1,1,-1]
Phi of K* [-2,-1,0,0,1,2,-1,1,1,1,3,0,1,0,1,0,1,1,1,1,0]
Phi of -K* [-2,-1,0,0,1,2,1,1,1,2,1,0,0,2,2,0,0,1,1,1,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+23t^4+14t^2+1
Outer characteristic polynomial t^7+33t^5+26t^3+4t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -256*K1**6 - 1472*K1**4*K2**2 + 3648*K1**4*K2 - 6080*K1**4 + 1536*K1**3*K2*K3 - 1344*K1**3*K3 - 448*K1**2*K2**4 + 2880*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 10688*K1**2*K2**2 - 1216*K1**2*K2*K4 + 11552*K1**2*K2 - 992*K1**2*K3**2 - 112*K1**2*K4**2 - 3972*K1**2 + 1024*K1*K2**3*K3 - 1600*K1*K2**2*K3 - 384*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 8416*K1*K2*K3 + 1280*K1*K3*K4 + 192*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1872*K2**4 - 64*K2**3*K6 - 656*K2**2*K3**2 - 128*K2**2*K4**2 + 1728*K2**2*K4 - 3564*K2**2 + 560*K2*K3*K5 + 80*K2*K4*K6 - 1736*K3**2 - 560*K4**2 - 132*K5**2 - 12*K6**2 + 4262
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{6}, {3, 5}, {2, 4}, {1}]]
If K is slice False
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