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Flat knot 6.954

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,2,0,2,1,1,1,1,1,0,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.954']
Arrow polynomial of the knot is: 12*K1**3 - 12*K1**2 - 8*K1*K2 - 5*K1 + 6*K2 + K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.954']
Outer characteristic polynomial of the knot is: t^7+27t^5+59t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.954']
2-strand cable arrow polynomial of the knot is: -448*K1**4*K2**2 + 800*K1**4*K2 - 2832*K1**4 + 128*K1**3*K2**3*K3 + 544*K1**3*K2*K3 - 384*K1**3*K3 - 2240*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 4544*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 11152*K1**2*K2**2 - 576*K1**2*K2*K4 + 10448*K1**2*K2 - 432*K1**2*K3**2 - 32*K1**2*K4**2 - 5084*K1**2 + 2944*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 224*K1*K2**2*K5 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7984*K1*K2*K3 + 720*K1*K3*K4 + 88*K1*K4*K5 - 224*K2**6 + 192*K2**4*K4 - 3456*K2**4 - 1184*K2**2*K3**2 - 96*K2**2*K4**2 + 2064*K2**2*K4 - 2222*K2**2 + 384*K2*K3*K5 + 24*K2*K4*K6 - 1640*K3**2 - 372*K4**2 - 52*K5**2 - 2*K6**2 + 4090
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.954']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4682', 'vk6.4981', 'vk6.6156', 'vk6.6633', 'vk6.8155', 'vk6.8567', 'vk6.9539', 'vk6.9886', 'vk6.20696', 'vk6.22136', 'vk6.28221', 'vk6.29646', 'vk6.39677', 'vk6.41918', 'vk6.46257', 'vk6.47864', 'vk6.48722', 'vk6.48941', 'vk6.49506', 'vk6.49711', 'vk6.50744', 'vk6.50953', 'vk6.51225', 'vk6.51418', 'vk6.57623', 'vk6.58781', 'vk6.62299', 'vk6.63232', 'vk6.67089', 'vk6.67953', 'vk6.69693', 'vk6.70376']
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 O1O2O3O4U5U1O6O5U6U4U2U3
R3 orbit {'O1O2O3O4U5U1O6O5U6U4U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U3U1U5O6O5U4U6
Gauss code of K* O1O2O3O4U5U3U4U2O6O5U1U6
Gauss code of -K* O1O2O3O4U5U4O6O5U3U1U2U6
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 0 2 1 0 -1],[ 2 0 1 2 0 2 -1],[ 0 -1 0 1 0 1 -1],[-2 -2 -1 0 0 -1 -1],[-1 0 0 0 0 0 -1],[ 0 -2 -1 1 0 0 -1],[ 1 1 1 1 1 1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -1 -1 -2],[-1 0 0 0 0 -1 0],[ 0 1 0 0 1 -1 -1],[ 0 1 0 -1 0 -1 -2],[ 1 1 1 1 1 0 1],[ 2 2 0 1 2 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,1,1,1,2,0,0,1,0,-1,1,1,1,2,-1]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,2,0,2,1,1,1,1,1,0,1,0,1,0]
Phi of -K [-2,-1,0,0,1,2,2,0,1,3,2,0,0,1,2,1,1,1,1,1,1]
Phi of K* [-2,-1,0,0,1,2,1,1,1,2,2,1,1,1,3,-1,0,0,0,1,2]
Phi of -K* [-2,-1,0,0,1,2,-1,1,2,0,2,1,1,1,1,1,0,1,0,1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2-4w^3z+24w^2z+29w
Inner characteristic polynomial t^6+17t^4+15t^2+1
Outer characteristic polynomial t^7+27t^5+59t^3+9t
Flat arrow polynomial 12*K1**3 - 12*K1**2 - 8*K1*K2 - 5*K1 + 6*K2 + K3 + 7
2-strand cable arrow polynomial -448*K1**4*K2**2 + 800*K1**4*K2 - 2832*K1**4 + 128*K1**3*K2**3*K3 + 544*K1**3*K2*K3 - 384*K1**3*K3 - 2240*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 4544*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 11152*K1**2*K2**2 - 576*K1**2*K2*K4 + 10448*K1**2*K2 - 432*K1**2*K3**2 - 32*K1**2*K4**2 - 5084*K1**2 + 2944*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 224*K1*K2**2*K5 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7984*K1*K2*K3 + 720*K1*K3*K4 + 88*K1*K4*K5 - 224*K2**6 + 192*K2**4*K4 - 3456*K2**4 - 1184*K2**2*K3**2 - 96*K2**2*K4**2 + 2064*K2**2*K4 - 2222*K2**2 + 384*K2*K3*K5 + 24*K2*K4*K6 - 1640*K3**2 - 372*K4**2 - 52*K5**2 - 2*K6**2 + 4090
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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