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Flat knot 6.1175

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,2,1,1,3,0,0,1,0,0,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1175', '7.24086']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.406', '6.410', '6.412', '6.1151', '6.1175', '6.1176']
Outer characteristic polynomial of the knot is: t^7+68t^5+122t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1175', '7.24086']
2-strand cable arrow polynomial of the knot is: -512*K1**6 + 1024*K1**4*K2**3 - 2944*K1**4*K2**2 + 3072*K1**4*K2 - 3104*K1**4 - 128*K1**3*K2**2*K3 + 1088*K1**3*K2*K3 - 352*K1**3*K3 + 1152*K1**2*K2**5 - 5248*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 6592*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 10224*K1**2*K2**2 - 768*K1**2*K2*K4 + 6744*K1**2*K2 - 96*K1**2*K3**2 - 32*K1**2*K4**2 - 1548*K1**2 + 256*K1*K2**5*K3 - 640*K1*K2**4*K3 - 256*K1*K2**4*K5 + 4032*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 2080*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 480*K1*K2**2*K5 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5280*K1*K2*K3 - 32*K1*K2*K4*K5 + 360*K1*K3*K4 + 88*K1*K4*K5 + 8*K1*K5*K6 - 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 - 3296*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 128*K2**3*K6 - 1008*K2**2*K3**2 - 32*K2**2*K3*K7 - 280*K2**2*K4**2 + 2024*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 + 94*K2**2 + 360*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K5*K7 - 632*K3**2 - 238*K4**2 - 60*K5**2 - 14*K6**2 + 1924
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1175']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.486', 'vk6.556', 'vk6.612', 'vk6.960', 'vk6.1057', 'vk6.1116', 'vk6.1651', 'vk6.1764', 'vk6.1829', 'vk6.2144', 'vk6.2241', 'vk6.2300', 'vk6.2579', 'vk6.2842', 'vk6.3056', 'vk6.3176', 'vk6.12055', 'vk6.13048', 'vk6.20485', 'vk6.20996', 'vk6.21840', 'vk6.22418', 'vk6.27882', 'vk6.28450', 'vk6.29392', 'vk6.32703', 'vk6.39323', 'vk6.40215', 'vk6.41503', 'vk6.46718', 'vk6.46858', 'vk6.57347']
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 O1O2O3O4U1U5U6O5O6U3U4U2
R3 orbit {'O1O2O3O4U1U5U6O5O6U3U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U1U2O5O6U5U6U4
Gauss code of K* O1O2O3U2U3O4O5O6U1U6U4U5
Gauss code of -K* O1O2O3U4U5O4O5O6U2U3U1U6
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 1 0 2 -1 1],[ 3 0 3 1 2 3 3],[-1 -3 0 -1 1 -2 0],[ 0 -1 1 0 1 -1 1],[-2 -2 -1 -1 0 -3 -1],[ 1 -3 2 1 3 0 1],[-1 -3 0 -1 1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 -1 -1 -1 -3 -2],[-1 1 0 0 -1 -1 -3],[-1 1 0 0 -1 -2 -3],[ 0 1 1 1 0 -1 -1],[ 1 3 1 2 1 0 -3],[ 3 2 3 3 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,1,1,1,3,2,0,1,1,3,1,2,3,1,1,3]
Phi over symmetry [-3,-1,0,1,1,2,-1,2,1,1,3,0,0,1,0,0,0,1,0,0,0]
Phi of -K [-3,-1,0,1,1,2,-1,2,1,1,3,0,0,1,0,0,0,1,0,0,0]
Phi of K* [-2,-1,-1,0,1,3,0,0,1,0,3,0,0,0,1,0,1,1,0,2,-1]
Phi of -K* [-3,-1,0,1,1,2,3,1,3,3,2,1,1,2,3,1,1,1,0,1,1]
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+52t^4+89t^2
Outer characteristic polynomial t^7+68t^5+122t^3+7t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -512*K1**6 + 1024*K1**4*K2**3 - 2944*K1**4*K2**2 + 3072*K1**4*K2 - 3104*K1**4 - 128*K1**3*K2**2*K3 + 1088*K1**3*K2*K3 - 352*K1**3*K3 + 1152*K1**2*K2**5 - 5248*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 6592*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 10224*K1**2*K2**2 - 768*K1**2*K2*K4 + 6744*K1**2*K2 - 96*K1**2*K3**2 - 32*K1**2*K4**2 - 1548*K1**2 + 256*K1*K2**5*K3 - 640*K1*K2**4*K3 - 256*K1*K2**4*K5 + 4032*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 2080*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 480*K1*K2**2*K5 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5280*K1*K2*K3 - 32*K1*K2*K4*K5 + 360*K1*K3*K4 + 88*K1*K4*K5 + 8*K1*K5*K6 - 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 - 3296*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 128*K2**3*K6 - 1008*K2**2*K3**2 - 32*K2**2*K3*K7 - 280*K2**2*K4**2 + 2024*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 + 94*K2**2 + 360*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K5*K7 - 632*K3**2 - 238*K4**2 - 60*K5**2 - 14*K6**2 + 1924
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {2, 3}, {1}]]
If K is slice False
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