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Flat knot 6.880

Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,1,1,2,3,3,0,0,0,0,1,0,1,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.880']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 14*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.880']
Outer characteristic polynomial of the knot is: t^7+43t^5+62t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.880']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1280*K1**4*K2**2 + 2176*K1**4*K2 - 3584*K1**4 - 256*K1**3*K2**2*K3 + 512*K1**3*K2*K3 + 32*K1**3*K3*K4 - 608*K1**3*K3 - 384*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2656*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 256*K1**2*K2**2*K4**2 + 672*K1**2*K2**2*K4 - 10064*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 192*K1**2*K2*K4**2 - 1536*K1**2*K2*K4 + 11464*K1**2*K2 - 448*K1**2*K3**2 - 32*K1**2*K3*K5 - 336*K1**2*K4**2 - 6868*K1**2 - 128*K1*K2**3*K3*K4 + 1632*K1*K2**3*K3 + 1024*K1*K2**2*K3*K4 - 1696*K1*K2**2*K3 + 384*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 1152*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 9288*K1*K2*K3 - 160*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2096*K1*K3*K4 + 416*K1*K4*K5 - 64*K2**6 - 64*K2**4*K3**2 - 64*K2**4*K4**2 + 544*K2**4*K4 - 2568*K2**4 + 192*K2**3*K3*K5 + 128*K2**3*K4*K6 - 64*K2**3*K6 - 1296*K2**2*K3**2 - 1032*K2**2*K4**2 + 2888*K2**2*K4 - 176*K2**2*K5**2 - 48*K2**2*K6**2 - 4542*K2**2 + 856*K2*K3*K5 + 312*K2*K4*K6 + 8*K2*K5*K7 - 2532*K3**2 - 1230*K4**2 - 136*K5**2 - 26*K6**2 + 5764
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.880']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4752', 'vk6.5079', 'vk6.6298', 'vk6.6737', 'vk6.8263', 'vk6.8712', 'vk6.9645', 'vk6.9960', 'vk6.20396', 'vk6.21743', 'vk6.27732', 'vk6.29272', 'vk6.39174', 'vk6.41404', 'vk6.45904', 'vk6.47543', 'vk6.48792', 'vk6.49003', 'vk6.49616', 'vk6.49819', 'vk6.50820', 'vk6.51035', 'vk6.51295', 'vk6.51490', 'vk6.57257', 'vk6.58472', 'vk6.61899', 'vk6.63010', 'vk6.66874', 'vk6.67746', 'vk6.69500', 'vk6.70220']
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 O1O2O3O4U1U5O6O5U4U6U2U3
R3 orbit {'O1O2O3O4U1U5O6O5U4U6U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U3U5U1O6O5U6U4
Gauss code of K* O1O2O3O4U5U3U4U1O5O6U2U6
Gauss code of -K* O1O2O3O4U5U3O5O6U4U1U2U6
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 -3 0 2 0 1 0],[ 3 0 2 3 1 3 1],[ 0 -2 0 1 -1 1 0],[-2 -3 -1 0 -1 -1 0],[ 0 -1 1 1 0 0 0],[-1 -3 -1 1 0 0 0],[ 0 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 0 0 0 -3],[-2 0 -1 0 -1 -1 -3],[-1 1 0 0 0 -1 -3],[ 0 0 0 0 0 0 -1],[ 0 1 0 0 0 1 -1],[ 0 1 1 0 -1 0 -2],[ 3 3 3 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,0,3,1,0,1,1,3,0,0,1,3,0,0,1,-1,1,2]
Phi over symmetry [-3,0,0,0,1,2,1,1,2,3,3,0,0,0,0,1,0,1,1,1,1]
Phi of -K [-3,0,0,0,1,2,1,2,2,1,2,0,1,0,1,0,1,2,1,1,0]
Phi of K* [-2,-1,0,0,0,3,0,1,1,2,2,0,1,1,1,-1,0,1,0,2,2]
Phi of -K* [-3,0,0,0,1,2,1,1,2,3,3,0,0,0,0,1,0,1,1,1,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+29t^4+37t^2+4
Outer characteristic polynomial t^7+43t^5+62t^3+11t
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 14*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial 256*K1**4*K2**3 - 1280*K1**4*K2**2 + 2176*K1**4*K2 - 3584*K1**4 - 256*K1**3*K2**2*K3 + 512*K1**3*K2*K3 + 32*K1**3*K3*K4 - 608*K1**3*K3 - 384*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2656*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 256*K1**2*K2**2*K4**2 + 672*K1**2*K2**2*K4 - 10064*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 192*K1**2*K2*K4**2 - 1536*K1**2*K2*K4 + 11464*K1**2*K2 - 448*K1**2*K3**2 - 32*K1**2*K3*K5 - 336*K1**2*K4**2 - 6868*K1**2 - 128*K1*K2**3*K3*K4 + 1632*K1*K2**3*K3 + 1024*K1*K2**2*K3*K4 - 1696*K1*K2**2*K3 + 384*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 1152*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 9288*K1*K2*K3 - 160*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2096*K1*K3*K4 + 416*K1*K4*K5 - 64*K2**6 - 64*K2**4*K3**2 - 64*K2**4*K4**2 + 544*K2**4*K4 - 2568*K2**4 + 192*K2**3*K3*K5 + 128*K2**3*K4*K6 - 64*K2**3*K6 - 1296*K2**2*K3**2 - 1032*K2**2*K4**2 + 2888*K2**2*K4 - 176*K2**2*K5**2 - 48*K2**2*K6**2 - 4542*K2**2 + 856*K2*K3*K5 + 312*K2*K4*K6 + 8*K2*K5*K7 - 2532*K3**2 - 1230*K4**2 - 136*K5**2 - 26*K6**2 + 5764
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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