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Flat knot 6.670

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,3,2,4,0,2,1,2,0,1,2,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.670']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 6*K1*K2 + 5*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.670', '6.768']
Outer characteristic polynomial of the knot is: t^7+69t^5+128t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.670']
2-strand cable arrow polynomial of the knot is: -64*K1**6 - 192*K1**4*K2**2 + 1344*K1**4*K2 - 3184*K1**4 + 288*K1**3*K2*K3 - 736*K1**3*K3 + 896*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 5728*K1**2*K2**2 - 544*K1**2*K2*K4 + 9136*K1**2*K2 - 752*K1**2*K3**2 - 64*K1**2*K4**2 - 6132*K1**2 + 160*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 192*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 8136*K1*K2*K3 - 32*K1*K2*K4*K5 + 1736*K1*K3*K4 + 200*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 888*K2**4 - 32*K2**3*K6 - 384*K2**2*K3**2 - 24*K2**2*K4**2 + 1616*K2**2*K4 - 5148*K2**2 + 536*K2*K3*K5 + 56*K2*K4*K6 - 2764*K3**2 - 934*K4**2 - 192*K5**2 - 28*K6**2 + 5348
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.670']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16571', 'vk6.16664', 'vk6.18125', 'vk6.18459', 'vk6.22974', 'vk6.23095', 'vk6.24584', 'vk6.24995', 'vk6.34963', 'vk6.35084', 'vk6.36723', 'vk6.37140', 'vk6.42536', 'vk6.42647', 'vk6.43995', 'vk6.44305', 'vk6.54818', 'vk6.54898', 'vk6.55943', 'vk6.56237', 'vk6.59250', 'vk6.59324', 'vk6.60481', 'vk6.60841', 'vk6.64792', 'vk6.64857', 'vk6.65604', 'vk6.65909', 'vk6.68094', 'vk6.68159', 'vk6.68679', 'vk6.68888']
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 O1O2O3O4U2O5U6U3O6U1U5U4
R3 orbit {'O1O2O3O4U2O5U6U3O6U1U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U4O6U2U6O5U3
Gauss code of K* O1O2O3U1U4U5U3O4U2O6O5U6
Gauss code of -K* O1O2O3U4O5O4U2O6U1U5U6U3
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 0 3 2 -1],[ 2 0 -1 2 4 2 1],[ 2 1 0 1 2 1 1],[ 0 -2 -1 0 1 0 0],[-3 -4 -2 -1 0 0 -3],[-2 -2 -1 0 0 0 -2],[ 1 -1 -1 0 3 2 0]]
Primitive based matrix [[ 0 3 2 0 -1 -2 -2],[-3 0 0 -1 -3 -2 -4],[-2 0 0 0 -2 -1 -2],[ 0 1 0 0 0 -1 -2],[ 1 3 2 0 0 -1 -1],[ 2 2 1 1 1 0 1],[ 2 4 2 2 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,1,2,2,0,1,3,2,4,0,2,1,2,0,1,2,1,1,-1]
Phi over symmetry [-3,-2,0,1,2,2,0,1,3,2,4,0,2,1,2,0,1,2,1,1,-1]
Phi of -K [-2,-2,-1,0,2,3,-1,0,1,3,3,0,0,2,1,1,1,1,2,2,1]
Phi of K* [-3,-2,0,1,2,2,1,2,1,1,3,2,1,2,3,1,0,1,0,0,-1]
Phi of -K* [-2,-2,-1,0,2,3,-1,1,2,2,4,1,1,1,2,0,2,3,0,1,0]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+47t^4+67t^2+1
Outer characteristic polynomial t^7+69t^5+128t^3+11t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 6*K1*K2 + 5*K2 + 2*K3 + 6
2-strand cable arrow polynomial -64*K1**6 - 192*K1**4*K2**2 + 1344*K1**4*K2 - 3184*K1**4 + 288*K1**3*K2*K3 - 736*K1**3*K3 + 896*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 5728*K1**2*K2**2 - 544*K1**2*K2*K4 + 9136*K1**2*K2 - 752*K1**2*K3**2 - 64*K1**2*K4**2 - 6132*K1**2 + 160*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 192*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 8136*K1*K2*K3 - 32*K1*K2*K4*K5 + 1736*K1*K3*K4 + 200*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 888*K2**4 - 32*K2**3*K6 - 384*K2**2*K3**2 - 24*K2**2*K4**2 + 1616*K2**2*K4 - 5148*K2**2 + 536*K2*K3*K5 + 56*K2*K4*K6 - 2764*K3**2 - 934*K4**2 - 192*K5**2 - 28*K6**2 + 5348
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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