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Flat knot 6.894

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,1,1,3,2,0,1,1,0,0,1,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.894', '7.28228']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.121', '6.125', '6.866', '6.894', '6.936', '6.937']
Outer characteristic polynomial of the knot is: t^7+38t^5+62t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.894', '7.28228']
2-strand cable arrow polynomial of the knot is: 1024*K1**4*K2**3 - 2560*K1**4*K2**2 + 2816*K1**4*K2 - 3776*K1**4 - 128*K1**3*K2**2*K3 + 768*K1**3*K2*K3 + 128*K1**3*K3*K4 - 320*K1**3*K3 + 1152*K1**2*K2**5 - 5248*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 6720*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 11504*K1**2*K2**2 - 640*K1**2*K2*K4 + 7680*K1**2*K2 - 512*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 1764*K1**2 + 256*K1*K2**5*K3 - 640*K1*K2**4*K3 - 256*K1*K2**4*K5 + 4032*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6352*K1*K2*K3 + 656*K1*K3*K4 + 120*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1600*K2**6 - 128*K2**5*K6 - 192*K2**4*K3**2 - 64*K2**4*K4**2 + 1408*K2**4*K4 - 3760*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 1232*K2**2*K3**2 - 32*K2**2*K3*K7 - 408*K2**2*K4**2 + 2320*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 + 46*K2**2 - 32*K2*K3**2*K4 + 472*K2*K3*K5 + 144*K2*K4*K6 + 8*K2*K5*K7 - 824*K3**2 - 306*K4**2 - 44*K5**2 - 6*K6**2 + 2264
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.894']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.46', 'vk6.93', 'vk6.190', 'vk6.247', 'vk6.282', 'vk6.657', 'vk6.664', 'vk6.1237', 'vk6.1328', 'vk6.1387', 'vk6.1428', 'vk6.1908', 'vk6.2368', 'vk6.2422', 'vk6.2621', 'vk6.2966', 'vk6.10082', 'vk6.10093', 'vk6.14593', 'vk6.15815', 'vk6.16216', 'vk6.17766', 'vk6.24266', 'vk6.29819', 'vk6.33407', 'vk6.33479', 'vk6.33555', 'vk6.36576', 'vk6.43686', 'vk6.53728', 'vk6.53778', 'vk6.63287']
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 O1O2O3O4U2U3O5O6U5U6U1U4
R3 orbit {'O1O2O3O4U2U3O5O6U5U6U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U4U5U6O5O6U2U3
Gauss code of K* O1O2O3O4U3U5U6U4O5O6U1U2
Gauss code of -K* O1O2O3O4U3U4O5O6U1U5U6U2
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 -2 0 3 -1 1],[ 1 0 -1 1 3 -1 1],[ 2 1 0 1 2 0 0],[ 0 -1 -1 0 1 0 0],[-3 -3 -2 -1 0 -1 1],[ 1 1 0 0 1 0 1],[-1 -1 0 0 -1 -1 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 1 -1 -1 -3 -2],[-1 -1 0 0 -1 -1 0],[ 0 1 0 0 0 -1 -1],[ 1 1 1 0 0 1 0],[ 1 3 1 1 -1 0 -1],[ 2 2 0 1 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,-1,1,1,3,2,0,1,1,0,0,1,1,-1,0,1]
Phi over symmetry [-3,-1,0,1,1,2,-1,1,1,3,2,0,1,1,0,0,1,1,-1,0,1]
Phi of -K [-2,-1,-1,0,1,3,0,1,1,3,3,1,0,1,1,1,1,3,1,2,3]
Phi of K* [-3,-1,0,1,1,2,3,2,1,3,3,1,1,1,3,0,1,1,-1,0,1]
Phi of -K* [-2,-1,-1,0,1,3,0,1,1,0,2,1,0,1,1,1,1,3,0,1,-1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 3z^2+16z+21
Enhanced Jones-Krushkal polynomial -2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w
Inner characteristic polynomial t^6+22t^4+31t^2
Outer characteristic polynomial t^7+38t^5+62t^3+7t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial 1024*K1**4*K2**3 - 2560*K1**4*K2**2 + 2816*K1**4*K2 - 3776*K1**4 - 128*K1**3*K2**2*K3 + 768*K1**3*K2*K3 + 128*K1**3*K3*K4 - 320*K1**3*K3 + 1152*K1**2*K2**5 - 5248*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 6720*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 11504*K1**2*K2**2 - 640*K1**2*K2*K4 + 7680*K1**2*K2 - 512*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 1764*K1**2 + 256*K1*K2**5*K3 - 640*K1*K2**4*K3 - 256*K1*K2**4*K5 + 4032*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6352*K1*K2*K3 + 656*K1*K3*K4 + 120*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1600*K2**6 - 128*K2**5*K6 - 192*K2**4*K3**2 - 64*K2**4*K4**2 + 1408*K2**4*K4 - 3760*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 1232*K2**2*K3**2 - 32*K2**2*K3*K7 - 408*K2**2*K4**2 + 2320*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 + 46*K2**2 - 32*K2*K3**2*K4 + 472*K2*K3*K5 + 144*K2*K4*K6 + 8*K2*K5*K7 - 824*K3**2 - 306*K4**2 - 44*K5**2 - 6*K6**2 + 2264
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}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {1, 3}, {2}]]
If K is slice False
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