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Flat knot 6.601

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,-1,1,2,2,4,1,1,0,2,1,1,1,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.601']
Arrow polynomial of the knot is: -6*K1**2 - 6*K1*K2 + 3*K1 + 3*K2 + 3*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.458', '6.601', '6.611', '6.995', '6.1026', '6.1037']
Outer characteristic polynomial of the knot is: t^7+59t^5+46t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.601', '6.672']
2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 64*K1**4*K2 - 288*K1**4 + 192*K1**3*K2*K3 - 224*K1**3*K3 + 352*K1**2*K2**3 - 3664*K1**2*K2**2 - 384*K1**2*K2*K4 + 4472*K1**2*K2 - 112*K1**2*K3**2 - 3996*K1**2 + 384*K1*K2**3*K3 - 544*K1*K2**2*K3 - 160*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 5408*K1*K2*K3 + 616*K1*K3*K4 + 104*K1*K4*K5 - 920*K2**4 - 448*K2**2*K3**2 - 56*K2**2*K4**2 + 1160*K2**2*K4 - 2894*K2**2 + 504*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 32*K3**2*K6 - 1860*K3**2 - 458*K4**2 - 144*K5**2 - 18*K6**2 + 3144
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.601']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71474', 'vk6.71501', 'vk6.71533', 'vk6.71560', 'vk6.72012', 'vk6.72031', 'vk6.72063', 'vk6.72083', 'vk6.72538', 'vk6.72545', 'vk6.72637', 'vk6.72662', 'vk6.72926', 'vk6.72965', 'vk6.73113', 'vk6.73140', 'vk6.73644', 'vk6.73679', 'vk6.73690', 'vk6.77103', 'vk6.77119', 'vk6.77155', 'vk6.77174', 'vk6.77452', 'vk6.77469', 'vk6.77943', 'vk6.77959', 'vk6.78579', 'vk6.81435', 'vk6.86901', 'vk6.87259', 'vk6.89350']
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 O1O2O3O4U2O5U3O6U5U1U6U4
R3 orbit {'O1O2O3O4U2O5U3O6U5U1U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U4U6O5U2O6U3
Gauss code of K* O1O2O3O4U2U5U6U4O5U1O6U3
Gauss code of -K* O1O2O3O4U2O5U4O6U1U5U6U3
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 -2 -1 3 0 2],[ 2 0 -1 0 4 1 2],[ 2 1 0 1 2 1 0],[ 1 0 -1 0 2 1 1],[-3 -4 -2 -2 0 -1 1],[ 0 -1 -1 -1 1 0 1],[-2 -2 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 2 0 -1 -2 -2],[-3 0 1 -1 -2 -2 -4],[-2 -1 0 -1 -1 0 -2],[ 0 1 1 0 -1 -1 -1],[ 1 2 1 1 0 -1 0],[ 2 2 0 1 1 0 1],[ 2 4 2 1 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,1,2,2,-1,1,2,2,4,1,1,0,2,1,1,1,1,0,-1]
Phi over symmetry [-3,-2,0,1,2,2,-1,1,2,2,4,1,1,0,2,1,1,1,1,0,-1]
Phi of -K [-2,-2,-1,0,2,3,-1,0,1,4,3,1,1,2,1,0,2,2,1,2,2]
Phi of K* [-3,-2,0,1,2,2,2,2,2,1,3,1,2,2,4,0,1,1,1,0,-1]
Phi of -K* [-2,-2,-1,0,2,3,-1,0,1,2,4,1,1,0,2,1,1,2,1,1,-1]
Symmetry type of based matrix c
u-polynomial -t^3+t^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+37t^4+25t^2+1
Outer characteristic polynomial t^7+59t^5+46t^3+5t
Flat arrow polynomial -6*K1**2 - 6*K1*K2 + 3*K1 + 3*K2 + 3*K3 + 4
2-strand cable arrow polynomial -64*K1**4*K2**2 + 64*K1**4*K2 - 288*K1**4 + 192*K1**3*K2*K3 - 224*K1**3*K3 + 352*K1**2*K2**3 - 3664*K1**2*K2**2 - 384*K1**2*K2*K4 + 4472*K1**2*K2 - 112*K1**2*K3**2 - 3996*K1**2 + 384*K1*K2**3*K3 - 544*K1*K2**2*K3 - 160*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 5408*K1*K2*K3 + 616*K1*K3*K4 + 104*K1*K4*K5 - 920*K2**4 - 448*K2**2*K3**2 - 56*K2**2*K4**2 + 1160*K2**2*K4 - 2894*K2**2 + 504*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 32*K3**2*K6 - 1860*K3**2 - 458*K4**2 - 144*K5**2 - 18*K6**2 + 3144
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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