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Flat knot 6.1376

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,1,2,2,3,1,0,1,2,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1376']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.238', '6.431', '6.945', '6.977', '6.981', '6.997', '6.1050', '6.1070', '6.1098', '6.1376']
Outer characteristic polynomial of the knot is: t^7+49t^5+90t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1348', '6.1374', '6.1376']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 512*K1**4*K2 - 1472*K1**4 + 256*K1**3*K2*K3 - 64*K1**3*K3 - 512*K1**2*K2**4 + 2624*K1**2*K2**3 - 7392*K1**2*K2**2 - 224*K1**2*K2*K4 + 6512*K1**2*K2 - 64*K1**2*K3**2 - 2640*K1**2 + 960*K1*K2**3*K3 - 1792*K1*K2**2*K3 - 256*K1*K2**2*K5 + 4400*K1*K2*K3 + 176*K1*K3*K4 + 16*K1*K4*K5 - 192*K2**6 + 256*K2**4*K4 - 2720*K2**4 - 96*K2**3*K6 - 384*K2**2*K3**2 - 24*K2**2*K4**2 + 1944*K2**2*K4 - 1062*K2**2 + 160*K2*K3*K5 + 24*K2*K4*K6 - 544*K3**2 - 188*K4**2 - 16*K5**2 - 2*K6**2 + 2058
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1376']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70512', 'vk6.70516', 'vk6.70577', 'vk6.70585', 'vk6.70730', 'vk6.70738', 'vk6.70829', 'vk6.70833', 'vk6.71089', 'vk6.71097', 'vk6.71220', 'vk6.71228', 'vk6.71289', 'vk6.71292', 'vk6.74728', 'vk6.74737', 'vk6.76241', 'vk6.76257', 'vk6.89181', 'vk6.89183']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3U4O5O6U1U5U6O4U2U3
R3 orbit {'O1O2O3U4O5O6U1U5U6O4U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U2O4U5U6U3O5O6U4
Gauss code of K* Same
Gauss code of -K* O1O2O3U1U2O4U5U6U3O5O6U4
Diagrammatic symmetry type +
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 -1 0 2],[ 3 0 2 3 1 1 2],[ 0 -2 0 1 0 -1 1],[-2 -3 -1 0 -2 -1 1],[ 1 -1 0 2 0 1 1],[ 0 -1 1 1 -1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 1 -1 -1 -2 -3],[-2 -1 0 -1 -1 -1 -2],[ 0 1 1 0 1 -1 -1],[ 0 1 1 -1 0 0 -2],[ 1 2 1 1 0 0 -1],[ 3 3 2 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,-1,1,1,2,3,1,1,1,2,-1,1,1,0,2,1]
Phi over symmetry [-3,-1,0,0,2,2,1,1,2,2,3,1,0,1,2,1,1,1,1,1,-1]
Phi of -K [-3,-1,0,0,2,2,1,1,2,2,3,1,0,1,2,1,1,1,1,1,-1]
Phi of K* [-2,-2,0,0,1,3,-1,1,1,2,3,1,1,1,2,-1,1,1,0,2,1]
Phi of -K* [-3,-1,0,0,2,2,1,1,2,2,3,1,0,1,2,1,1,1,1,1,-1]
Symmetry type of based matrix +
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+31t^4+52t^2+4
Outer characteristic polynomial t^7+49t^5+90t^3+8t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -256*K1**4*K2**2 + 512*K1**4*K2 - 1472*K1**4 + 256*K1**3*K2*K3 - 64*K1**3*K3 - 512*K1**2*K2**4 + 2624*K1**2*K2**3 - 7392*K1**2*K2**2 - 224*K1**2*K2*K4 + 6512*K1**2*K2 - 64*K1**2*K3**2 - 2640*K1**2 + 960*K1*K2**3*K3 - 1792*K1*K2**2*K3 - 256*K1*K2**2*K5 + 4400*K1*K2*K3 + 176*K1*K3*K4 + 16*K1*K4*K5 - 192*K2**6 + 256*K2**4*K4 - 2720*K2**4 - 96*K2**3*K6 - 384*K2**2*K3**2 - 24*K2**2*K4**2 + 1944*K2**2*K4 - 1062*K2**2 + 160*K2*K3*K5 + 24*K2*K4*K6 - 544*K3**2 - 188*K4**2 - 16*K5**2 - 2*K6**2 + 2058
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}]]
If K is slice False
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